Time Discretization Schemes for Poincaré Waves in Finite-Element Shallow-Water Models

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

SIAM J. SCI. COMPUT. c© 2011 Society for Industrial and Applied MathematicsVol. 33, No. 5, pp. 2217–2246

TIME DISCRETIZATION SCHEMES FOR POINCARE WAVES INFINITE-ELEMENT SHALLOW-WATER MODELS∗

DANIEL Y. LE ROUX† , MICHEL DIEME‡ , AND ABDOU SENE§

Abstract. The finite-element spatial discretization of the linear shallow-water equations isexamined in the context of several temporal discretization schemes. Three finite-element pairs areconsidered, namely, the P0 − P1, P

NC1 − P1, and RT0 − P0 schemes, and the backward and forward

Euler, Crank–Nicolson, and second and third order Adams–Bashforth time stepping schemes areemployed. A Fourier analysis is performed at the discrete level for the Poincare waves, and itdetermines the stability limit of the schemes and the error in wave amplitude and phase that can beexpected. Numerical solutions of test problems to simulate Poincare waves illustrate the analyticalresults.

Key words. shallow water equations, finite-element method, time discretization, stability ofnumerical methods, dispersion analysis, Poincare waves

AMS subject classifications. 35L99, 74S05, 65M99, 65M12, 65T99, 74J15

DOI. 10.1137/090779413

1. Introduction. The shallow-water (SW) model is derived from the Navier–Stokes equations by vertical integration under Boussinesq and hydrostatic pressureassumptions. This simplified set of equations retains much of the dynamical com-plexity of three-dimensional flows on the rotating Earth and it can represent severalclasses of wave motions: external inertia-gravity (Poincare), Kelvin, and large-scaleplanetary (Rossby) waves [10, 33]. As a prototype of the primitive equations, theSW model has been employed to test numerical schemes for a variety of problems ofcoastal and environmental engineering, including oceanic, atmospheric, and ground-water flows [2, 8, 11, 20, 31, 41, 43]. A number of theoretical and numerical resultshave been obtained for these problems in the past few decades.

As for the primitive system, the discrete SW model usually suffers from spurioussolutions that arise due to the coupling between the momentum and continuity equa-tions. The spurious modes usually take the form of surface-elevation, velocity, and/orCoriolis modes. They are small-scale artifacts introduced by the spatial discretiza-tion scheme which do not propagate but are trapped within the model grid, leadingto noisy solutions. Their appearance is encountered in most finite-difference (FD)[1, 5, 6, 34, 44] and Galerkin formulations [27, 28, 36, 37, 39, 40, 42] and is mainlydue to an inappropriate placement of variables on the grid and/or a bad choice ofapproximation function spaces.

In the past few years, the Galerkin-type methods have appeared to be promisingalternatives to classical FD schemes for river, costal, and ocean modeling. Indeed, tri-

∗Submitted to the journal’s Methods and Algorithms for Scientific Computing section December 8,2009; accepted for publication (in revised form) May 2, 2011; published electronically September 13,2011. This work is supported by grants from the Natural Sciences and Engineering Research Council(NSERC) and ArcticNet.

http://www.siam.org/journals/sisc/33-5/77941.html†CNRS, Universite Lyon 1, Institut Camille Jordan, 43, blvd du 11 novembre 1918, 69622 Villeur-

banne Cedex, France ([email protected]).‡Departement de Mathematiques et de Statistique, Universite Laval, Quebec, QC, G1K 7P4

Canada ([email protected]).§UFR des sciences Appliquees et Technologie, Universite Gaston Berger, BP 234, Saint Louis,

Senegal ([email protected]).

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2218 DANIEL Y. LE ROUX, MICHEL DIEME, AND ABDOU SENE

angular elements offer the enhanced flexibility of using grids of variable sizes, shapes,and orientation for representing the boundaries of complex domains and a naturaltreatment of boundary conditions. In order to select appropriate spatial discretiza-tion schemes, dispersion and kernel analyses have been performed to ascertain thepresence and determine the form of spurious modes as well as the dispersive natureof the Galerkin formulation of the linearized SW equations.

Such analyses have been conducted on both uniform [3, 12, 13, 16, 24, 25, 27, 28,32, 42] and unstructured meshes [7, 14, 16, 36]. They illustrate how phase and groupvelocity can help in the selection of a spatial discretization scheme. In particular, theP0 − P1, P

NC1 − P1, and RT0 − P0 finite-element (FE) pairs have been identified as

a promising compromise for the discretization of the inviscid linear SW equations,provided the grid resolution is high relative to the Rossby radius of deformation forthe RT0 − P0 scheme [27].

The problem of accuracy in propagating inertia-gravity waves on two-dimensional(2D) FD grids is investigated in [6]. It is shown that the sole analysis of the spatialdiscretization has to be reanalyzed when time discretization is taken into accountas well. Most of the above-mentioned analyses have been performed by assumingthat time is continuous, except in [25] and [32], where the one-dimensional (1D)SW equations are solved using the discontinuous Galerkin and upwind finite-volumeschemes, respectively. In [16], semi- and fully implict time integrations are performedfor the 2D SW model using the PNC

1 −P1 and RT0−P0 FE pairs, but the amplitudeand phase speed of the wave are only numerically computed on arbitrary meshes. Inthe present study, a Fourier analysis is conducted for the space and time variables inthe case of the inertia-gravity waves. The FE spatial discretization of the 2D linearSW equations employing the P0 − P1, P

NC1 − P1, and RT0 − P0 pairs is re-examined

(compared to [27] where time is assumed continuous) in the context of five temporaldiscretization schemes. The latter are the backward Euler (EB), forward Euler (EF),Crank–Nicolson (CN), and second (AB2) and third (AB3) order Adams–Bashforthschemes. Our goal is to determine the stability limit of the schemes and the error inwave amplitude and phase that can be expected for the largest allowable time-step.This work can also be considered as an extension of [15] where the 1D SW model isanalyzed using linear finite elements for all variables and the leap frog and AB2 timediscretization schemes.

The reason for employing temporal explicit schemes is the following. In [29] it isfound that the P0 −P1, P

NC1 −P1, and RT0 −P0 FE schemes can be advantageously

lumped in SW simulations for the propagation of gravity and Rossby waves, withoutsacrificing the model’s accuracy and dispersion properties, as long as the mesh reso-lution is not too coarse. Only the modes corresponding to the smallest wavelengthsare slowed down by the numerical schemes while the others remain mostly unaffected.In the purpose of using a mass lumping strategy, the explicit time stepping schemesEF, AB2, and AB3 are employed in this study. Indeed, the use of explicit temporalschemes combined with mass lumping would allow us to compute the SW solutionwithout solving a linear system. The resulting model would then combine the advan-tages of both fast and simple FD schemes and unstructured and flexible FE ones.

The paper is organized as follows. The linear SW equations and the exact freesolutions are presented in section 2. The discrete model, including the time and spacediscretization procedures, is obtained in section 3. The computation of the discretedispersion relations is performed in section 4 and the results are analyzed in section 5.The results of test experiments are presented in section 6 and some concluding remarkscomplete the study.

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TEMPORAL SCHEMES FOR POINCARE WAVES IN FE SW MODELS 2219

2. The continuous model and its exact free solutions. Let Ω be the modeldomain with boundary Γ. In this study, attention will be restricted to the inviscid, lin-earized about a state of rest, depth averaged, SW equations with constant bathymetry.Indeed, these equations are satisfactory for studying the stability properties of timeand spatial discretization methods. In Cartesian coordinates they are expressed [21]as

ut + f k× u+ g∇η + λu = 0 ,(2.1)

ηt +H ∇ · u = 0 ,(2.2)

where u(x, t) = (u, v) is the velocity field with x = (x, y), η(x, t) is the surface eleva-tion with respect to the reference level z = 0, g and f are the gravitational accelerationand Coriolis parameter, respectively, k = (0, 0, 1) is a unit vector in the vertical di-rection leading to k × u = (−v, u) by considering that the vertical component of uis zero, λ is the linear bottom friction coefficient, and the mean depth H is assumedconstant. Note that η would be the pressure in the Navier–Stokes equations. Thegoverning equations (2.1) and (2.2) describe a first order hyperbolic system. Initialconditions and periodic boundary conditions complete the mathematical statement ofthe problem.

The free modes of (2.1)–(2.2) are examined by perturbing about the basic stateu = v = η = 0. Because the governing equations are linear, the solution may beexamined by considering the behavior of one Fourier mode. We then seek solutionsof (2.1)–(2.2) of the form (u, v, η) = (u, v, η) ei(kx+ly+ωt), where k and l are the wavenumbers in the x- and y-directions, respectively, and ω is the harmonic frequency.Substitution into (2.1)–(2.2) leads to the 3× 3 matrix system for the amplitudes

(2.3)

⎛⎝ iω + λ −f igkf iω + λ igliHk iHl iω

⎞⎠⎛⎝ uvη

⎞⎠ = 0.

For a nontrivial solution to exist, the determinant of the matrix in the left-hand side(LHS) of (2.3) must equal zero, and this yields the so-called dispersion relation

(2.4) ω3 − 2iλω2 −[gH(k2 + l2) + f2 + λ2

]ω + igHλ(k2 + l2) = 0.

Solving (2.4) leads to three solutions for the frequency. One is the geostrophic mode,and it would correspond to the slow Rossby mode on a β-plane, while the other twosolutions correspond to the free-surface gravitational modes with rotational correction.

3. The discrete model. The discretization of (2.1) and (2.2) is now performedin time and space in sections 3.1 and 3.2, respectively. Five time stepping schemesare used in section 3.1. These are the EB, EF, CN, AB2, and AB3 schemes. The FEmethod is employed in section 3.2 with the P0 − P1, P

NC1 − P1, and RT0 − P0 pairs.

3.1. Time discretization. For a given time step Δt = tn+1−tn, n = 0, 1, 2, . . . ,with tn = nΔt, we introduce a four-level time discretization of (2.1)–(2.2) of the form

un+1 − un

Δt+ f k×

(α1u

n+1 + α2un + α3u

n−1 + α4un−2)

(3.1)

+ g(α1∇η n+1 + α2∇η n + α3∇η n−1 + α4∇η n−2

)+ λ

(α1u

n+1 + α2un + α3u

n−1 + α4un−2)= 0,

η n+1 − η n

Δt+H

(α1∇.un+1 + α2∇.un + α3∇.un−1 + α4∇.un−2

)= 0,(3.2)

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where αi, i = 1, 2, 3, 4, are real parameters with∑3

i=1 αi = 1. The standard choice forαi, leading to the EB, EF, CN, AB2, and AB3 schemes, is summarized in Table 3.1.

Table 3.1

Values of the parameters αi, i = 1, 2, 3, 4, in (3.1)–(3.2), leading to the EB, EF, CN, AB2, andAB3 time stepping schemes.

EB EF CN AB2 AB3

α1 1 0 12

0 0

α2 0 1 12

32

2312

α3 0 0 0 − 12

− 43

α4 0 0 0 0 512

Because (3.1) and (3.2) are linear equations with constant coefficients, we seek pe-riodic Fourier solutions of the form un = u(x) eiωtn , η n = η(x) eiωtn , n = 1, 2, 3, . . . ,where u(x) and η(x) are the amplitudes of the velocity field and surface elevation,respectively, with tn = nΔt. By inserting the Fourier expansions into (3.1) and (3.2)and letting E = eiωΔt we obtain

a3 u+ f Δt a2 k× u+ gΔt a2 ∇η = 0,(3.3)

a1 η +H Δt a2 ∇ · u = 0,(3.4)

where a1 = E − 1, a2 = α1E + α2 + α3E−1 + α4E

−2, and a3 = a1 + λΔt a2.

We now perform the spatial discretization using the Galerkin FE method.

3.2. Spatial discretization. We first introduce the weak formulation and thendescribe the three FE pairs that are employed in this study. Finally, the Galerkin FEdiscretization using these pairs is presented.

3.2.1. The weak formulation. Two weak formulations are proposed in whatfollows to correspond with the three FE pairs mentioned above. The Sobolev spaceH1 (Ω) denotes the space of functions in the square-integrable space L2 (Ω) whosefirst derivatives belong to L2 (Ω).

Formulation 1. Let η be in a subspace G of H1(Ω), and let each component ofthe velocity field belong to a subspace V of L2(Ω). We multiply (3.3) and (3.4) bytest functions ϕ(x) (whose x- or y-component is formally denoted by ϕ) and ψ(x)belonging to V 2 and G, respectively, and we integrate over the domain Ω to obtain

a3

∫Ω

u ·ϕ dx+ f Δt a2

∫Ω

(k× u) · ϕ dx+ gΔt a2

∫Ω

∇η · ϕ dx = 0,(3.5)

a1

∫Ω

η ψ dx−H Δt a2

∫Ω

u · ∇ψ dx = 0,(3.6)

by applying the periodic boundary conditions used in this paper in (3.6).

Formulation 2. Let u belong to W, a subspace of H(div,Ω), and η belong toQ, a subspace of L2(Ω). The functional space H(div,Ω) is the space of functionsbelonging to (L2(Ω))2 whose divergence belongs to L2(Ω). The weak formulation isobtained by multiplying (3.3) and (3.4) by test functions φ(x) and ξ(x) belonging toW and Q, respectively, and by integrating over the domain and applying the boundary

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conditions in (3.7)

a3

∫Ω

u · φ dx + f Δt a2

∫Ω

(k× u) · φ dx− gΔt a2

∫Ω

η∇ · φ dx = 0,(3.7)

a1

∫Ω

η ξ dx+HΔt a2

∫Ω

∇ · u ξ dx = 0.(3.8)

3.2.2. The FE pairs. We now describe the three FE pairs which are employedin this study, namely, the P0 − P1, P

NC1 − P1, and RT0 − P0 pairs. Conventional FE

terminology is adopted and the nomenclature Pm−Pn means that velocity componentsand surface elevation are, respectively, represented as piecewise-defined polynomialsof degree m and n. Common to the P0 −P1 and PNC

1 −P1 pairs is a piecewise-linearcontinuous representation of surface elevation, and they differ from one another intheir representation of velocity. The latter is piecewise-constant for the P0 − P1 pairand PNC

1 for the PNC1 − P1 pair.

(a) (b) (c)

K1

K2

A B

CD1

2 3 4

5

3

A

C

B

2

1

4

5

DK

K

1

2

K1

K2

A B

CD

n

nn n

n

1

2 3 4

5

Fig. 3.1. Compact supports of the PNC1 (a)–(b) and RT0 (c) basis functions at node 3. The

PNC1 linear basis function at node 3 is in bold (b). Velocity and surface-elevation node locations are

represented by the symbols • and ©, respectively, for the PNC1 −P1 (a)–(b) and RT0 −P0 (c) pairs.

The PNC1 FE [9, 17, 19, 26, 38] has velocity nodes at triangle midedge points, and

those are represented by the symbol • in Figure 3.1(a)–(b). Linear basis functionsare used to approximate the two velocity components on the element’s two-trianglesupport. For example, the compact support of the PNC

1 linear basis function atnode 3 is made up of triangles K1 = (A,D,C) and K2 = (A,C,B). As shown inFigure 3.1(b), the PNC

1 basis function (in bold) is zero outside the triangles ADCand ACB and discontinuous along sides AB, BC, AD, and DC. It takes the values1 at node 3 and along the triangle side AC, 0 at velocity nodes 1, 2, 4, 5, and −1at surface-elevation nodes B and D. Since this particular representation of velocityis continuous only across triangle boundaries at midedge points and discontinuouseverywhere else around a triangle boundary, this element is termed nonconforming(NC) in the FE literature. A useful property of the PNC

1 element is that the velocityand Coriolis mass matrices are diagonal, due to the orthogonality property of thePNC1 basis functions [38]. Such a desirable and unusual property of the FE method

greatly enhances computational efficiency. Piecewise-linear continuous basis functions(P1) are used to approximate the surface elevation at triangle vertices represented bythe symbol © in Figure 3.1(a)–(b). If ϕ(x) and ψ(x) represent the PNC

1 and P1 basisfunctions, respectively, we have ϕp = 1 − 2ψs, over a given triangle K, if p is themidedge velocity node of K facing the surface elevation vertex s of K.

The RT0 − P0 pair, also called the lowest-order Raviart–Thomas element [35], isbased on flux conservation on element edges and has normal velocity components at

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triangle midedge points and a piecewise-constant representation of surface elevation.The velocity RT0 basis functions, denoted by φ(x), are piecewise linear, and they aredefined with respect to the orientation of the chosen normal vector n to the faces. Forinstance, at node 3 in Figure 3.1(c), we have φ3(x) = (x − xD)/2A(K1) if x ∈ K1,φ3(x) = −(x − xB)/2A(K2) if x ∈ K2, and 0 otherwise, where xD and xB are thecoordinates of D and B, respectively, and A(Kj) is the area of triangle Kj, j = 1, 2.

3.2.3. Galerkin FE discretizations. The Galerkin method approximates thesolutions of (3.5)–(3.6) and (3.7)–(3.8) in finite-dimensional subspaces. Consider a FEtriangulation Th, of the polygonal domain Ω, where h is a representative meshlengthparameter that measures resolution. For the purposes of the following analysis, itsuffices that we consider a uniform mesh made up of biased right triangles as inFigure 3.1, and h is thus taken as a constant in the x- and y-directions.

For the P0 − P1 and PNC1 − P1 pairs let Vh and Gh be the finite-dimensional

subspaces of V and G, respectively. The solution uh belongs to Vh × Vh definedto be the set of functions uh whose restriction on a triangle K of Th belongs toPm(K) × Pm(K), where Pm(K) denotes the set of polynomials of degree m definedon K. We have m = 0 for the P0 − P1 pair and m = 1 for the PNC

1 − P1 pair. Forthe latter pair, uh is continuous only at the midpoint of each face of Th. The solutionηh belongs to Gh defined to be the set of functions ηh whose restriction on K of Thbelongs to P1(K).

We then expand uh = (uh, vh) and ηh over K with respect to the basis functionsϕi and ψj belonging to Vh and Gh, respectively, uh =

∑i∈IK

uiϕi, ηh =∑

j∈JKηjψj ,

where IK denotes the set of velocity nodes (barycenters and midside nodes for theP0−P1 and P

NC1 −P1 pairs, respectively), and JK is the set of surface-elevation nodes

(vertices) of K. We replace ϕ ∈ V × V and ψ ∈ G by the corresponding FE testfunctions ϕh ∈ Vh × Vh and ψh ∈ Gh in (3.5) and (3.6), respectively, and we obtain

a3∑

K∈Th

∫K

uh · ϕh dx+Δt a2∑

K∈Th

∫K

(f (k× uh) ·ϕh + g∇ηh ·ϕh

)dx = 0,(3.9)

a1∑

K∈Th

∫K

ηh ψh dx−H Δt a2∑

K∈Th

∫K

uh · ∇ψh dx = 0.(3.10)

For the RT0 −P0 pair, let Wh and Qh be the finite-dimensional subspaces of W andQ, respectively. The solution uh belongs to Wh defined to be the set of functionsuh whose restriction on a triangle K of Th belongs to the Raviart–Thomas vector FEspace of lowest order

RT0(K) = P0(K)× P0(K) + xP0(K) = {uh = (a, b)t + cxt, a, b, c ∈ R}.

For a midpoint node i located at face ei, a typical nodal value of the RT0 velocityfield at node i is defined as Ji =

∫eiuh · n de, and Ji is well defined since uh · n is

continuous at the face ei.The solution ηh belongs to Qh defined to be the set of functions whose restriction

on K of Th belongs to P0(K), and the nodal values are located at the barycenter ofeach triangle K. The discrete velocity field and surface elevation are expanded overeach triangle K with respect to the basis functions φi and ξj belonging to Wh andQh, respectively, uh =

∑i∈IK

Jiφi, ηh =∑

k∈BKηkξk, where IK and BK denote

the set of midside nodes and barycenters of K, respectively. By replacing φ ∈ W

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and ξ ∈ Q by the corresponding FE test functions φh ∈ Wh and ξh ∈ Qh in (3.7)and (3.8), respectively, we obtain

a3∑

K∈Th

∫K

uh · φh dx+Δt a2∑

K∈Th

∫K

(f (k× uh) · φh − g ηh ∇ · φh

)dx = 0,(3.11)

a1∑

K∈Th

∫K

ηh ξh dx+H Δt a2∑

K∈Th

∫K

∇ · uh ξh dx = 0.(3.12)

We now compute the discrete dispersion relations for the three FE pairs.

4. Computation of the discrete dispersion relations. The discrete disper-sion relations are obtained for the P0−P1, P

NC1 −P1, and RT0−P0 pairs in sections 4.2,

4.3, and 4.4, respectively, and the dispersion analysis is performed in section 5.

4.1. Introduction. For the purposes of the following analysis, we consider auniform mesh made up of biased right triangles, as previously mentioned, and h isthus taken as a constant in the x- and y- directions. In order to compute the dispersionrelation we follow the same procedure as in section 2 for the continuous case, but atthe discrete level.

(a) P0 − P1 (b) PNC1 − P1 (c) RT0 − P0

C

C2

1

S

S

V

H

D

H

VD

C1

C2

Fig. 4.1. Velocity and surface-elevation node locations are represented by the symbols • and ©,respectively, for the (a) P0 −P1, (b) PNC

1 −P1, and (c) RT0 − P0 pairs. The black arrows indicatethe location of normal velocity nodes, and the arrow points in the direction of the chosen normal.The capital letters refer to the typical velocity and elevation nodes used in the present section.

In [3, 12, 13], common linear interpolation is considered for both the velocity andpressure variables (the so-called P1 − P1 pair). In that particular case, the Fourieranalysis is straightforward as all the discrete velocity and pressure unknowns arelocated on triangle vertices. It is hence sufficient to consider one discrete equation(for momentum and continuity) at a typical vertex node for the Fourier analysis, dueto symmetry reasons.

In this paper the situation is quite different since each of the three FE pairs has adifferent approximation for velocity and pressure variables, and velocity and pressurenodal locations are different. In fact, nodal unknowns may be located on typical nodalsets, vertices, faces, and barycenters, and selected discrete equations at these nodesneed to be retained for the Fourier analysis. Note that the nodal unknowns belongingto the same set are distributed on a regular grid of size h. For example, three discretemomentum equations are considered for the PNC

1 − P1 (on the x- and y-axes) andRT0 −P0 (for the normal velocity) pairs on the three possible types of faces, denotedby H (horizontal), V (vertical), and D (diagonal), shown in Figure 4.1(b) and (c).Only one discrete continuity equation is retained at a typical vertex node (S), shownin Figure 4.1(a) and (b), for the P0 − P1 and PNC

1 − P1 pairs. Finally, two discreteequations are considered at the two possible types of barycenters corresponding to

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(a) P0 − P1 (b) PNC1 − P1 (c) RT0 − P0

F

BA

G

E

D C

K1

K2

K3

K4

K5

K6 2

6

11108 9

4 53

1

12

7

CD

F G

E BA

1 2

6

15

73

1410

16

5

9

11 1312

8

4

G H

AF

I

B

CDE

K1

K2

K3

K4

K5

K6

K7

K8

Fig. 4.2. Velocity and surface-elevation nodal locations employed to compute the selected dis-crete equations for the (a) P0 − P1, (b) PNC

1 − P1, and (c) RT0 − P0 pairs.

upward (C1) and downward (C2) pointing triangles, shown in Figure 4.1(a) and (c),for the P0 − P1 (velocity) and RT0 − P0 (elevation) pairs. The selected discreteequations are obtained for each FE pair by using the velocity and surface-elevationnodal locations shown in Figure 4.2.

As for the continuum case, the dispersion relations for the discrete schemesare found through a Fourier expansion. The discrete solutions corresponding to(uj , vj , ηj) = (u, v, η) ei(kxj+lyj) are sought at node j, as in Figure 4.2, where (uj , vj , ηj)are the nodal unknowns that appear in the selected discrete equations and (u, v, η)are the amplitudes. For the RT0 − P0 pair, the nodal values of the velocity field

corresponding to Jj = J ei(kxj+lyj) are sought at node j, where J is the amplitude.The (xj , yj) coordinates are expressed in terms of a distance to a reference node. Forthe three FE pairs that are considered in this study, substitution of (uj , vj , ηj) or(Jj , ηj) in the discrete equations leads to a matrix system for the amplitudes afterlong and tedious algebra, and only the result is given here. The dispersion relationis then obtained by setting the determinant of the matrix system to zero. We obtaina polynomial of degree nr, ranging from 5 to 21, depending on the choice of the FEpair and time stepping scheme.

4.2. The P0 −P1 pair. The spatially discrete operators in (3.9) and (3.10) areobtained from the stencils in [27, Figures 3.1 and 3.3]. The velocity and elevationnode locations are shown in Figure 4.2(a).

At the surface-elevation node A, for ψh = ψA, (3.10) leads to

a1 (6 ηA + ηB + ηC + ηD + ηE + ηF + ηG)

− 6Ha2Δt

h(uK4

+ uK3− uK6

− uK1+ vK4

+ vK5− vK1

− vK2) = 0.(4.1)

Equation (3.9) is rewritten in the x- (with ϕh = (ϕh, 0)) and y- (with ϕh = (0, ϕh))directions. The discrete momentum equations are then evaluated at velocity nodesK1 andK2 (identified with trianglesK1 andK2, respectively), i.e., at the two possibletypes of barycenters corresponding to upward and downward pointing triangles. TheP0 test function ϕh is replaced by the characteristic function on triangle Ki, which isdefined to have the value 1 on Ki and zero elsewhere, and uh is expanded over each

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triangle. We obtain four discrete scalar momentum equations at nodes K1 and K2,

a3uK1+ f Δt a2 k× uK1

+ ga2Δt

hGK1

= 0,(4.2)

a3uK2+ f Δt a2 k× uK2

+ ga2Δt

hGK2

= 0,(4.3)

where GK1= (ηB − ηA, ηC − ηA)

t and GK2= (ηC − ηD, ηC − ηA)

t.As mentioned above, two possible types of barycenters, i.e., upward and downward

pointing triangles (C1 and C2, respectively), and only one typical vertex node (S),as indicated in Figure 4.1(a), have to be considered. This leads us to distinguishtwo amplitudes for the velocity field denoted by uC1

and uC2and only one amplitude

named ηS for the surface-elevation. Substitution of the Fourier mode into the discretecontinuity equation (4.1) and the four discrete scalar momentum equations (4.2)–(4.3)leads to the following 5× 5 square matrix system for the Fourier amplitudes:

(4.4)

⎛⎜⎜⎝M 0

0 M−gΔta2D

t

HΔta2D 13aa1

⎞⎟⎟⎠⎛⎜⎜⎝

uC1

uC2

ηS

⎞⎟⎟⎠ = 0,

where M =(

a3 −fΔta2fΔta2 a3

), a = 3+ cos(kh) + cos(lh) + cos(kh− lh), and

D =(−d1,1 −d1,2 d1,1 d1,2

), with

d1,1 = i 2h sin kh2 ei(k−2l) h

6 , d1,2 = i 2h sin lh2 e

i(l−2k) h6 .

For a nontrivial solution (uC1, uC2

, ηS) to exist, the 5× 5 determinant of the matrix

in the LHS of (4.4) must vanish. This condition leads to a polynomial of degree nrin E (the so-called dispersion relation), and nr is shown in Table 4.1.

Table 4.1

Degree nr of the dispersion relations for the EB, EF, CN, AB2, and AB3 time stepping schemesand the P0 − P1, P

NC1 − P1, and RT0 − P0 pairs. For the continuous case we have nr = 3.

EB EF CN AB2 AB3

P0 − P1 5 5 5 10 15

PNC1 − P1 7 7 7 14 21

RT0 − P0 5 5 5 10 15

4.3. The PNC1 − P1 pair. We again employ (3.9) and (3.10) to compute the

discrete dispersion relation. The spatially discrete operators are obtained from thestencils in [27, Figures 3.1 and 3.5].

The velocity and surface-elevation node locations are shown in Figure 4.2(b). Atthe surface-elevation node A, for ψh = ψA, (3.10) leads to

a1 (6 ηA + ηB + ηC + ηD + ηE + ηF + ηG)

− 2H a2Δt

h(u2 + u3 + u9 + u8 + 2 u6 − u4 − u5 − u10 − u11 − 2 u7)

− 2H a2Δt

h(v6 + v2 + v4 + v1 + 2 v3 − v7 − v11 − v9 − v12 − 2 v10) = 0.(4.5)

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In order to compute the discrete momentum equations at nodes 3, 4, and 6, lying onvertical, diagonal, and horizontal faces, respectively, (3.9) is rewritten in the x- (withϕh = (ϕh, 0)) and y- (with ϕh = (0, ϕh)) directions and evaluated at velocity nodes3, 4, and 6 by replacing the PNC

1 test function ϕh by ϕi, i = 3, 4, 6, and expandinguh and ηh over each triangle of the compact support of ϕi, i = 3, 4, 6. We then obtainthe six following discrete scalar equations at nodes 3, 4, and 6:

a3 u3 + f Δt a2 k× u3 +gΔt

2ha2 G3 = 0,(4.6)


2ha2 G4 = 0,(4.7)


2ha2 G6 = 0,(4.8)

where G3 = (ηG − ηF + ηA − ηE , 2 ηA − 2 ηF )t, G4 = (ηG − ηF + ηB − ηA, ηA − ηF +

ηB − ηG)t, and G6 = (2 ηA − 2 ηE , ηA − ηF + ηD − ηE)

t.

Note that, as already mentioned, we need to consider three possible types of faces,i.e., horizontal, vertical, and diagonal (written as H , V , and D, respectively) and onlyone typical vertex node (written as S), as indicated in Figure 4.1(b). This leads usto distinguish three amplitudes for the velocity field denoted by uH , uV , and uD andonly one amplitude named ηS for the surface-elevation.

Substitution of the Fourier mode into the discrete continuity equation (4.5) andthe six discrete scalar momentum equations (4.6)–(4.8) leads to the following 7 × 7square matrix system for the Fourier amplitudes:

(4.9)

⎛⎜⎜⎝M 0 0

0 M 0 −gΔta2Dt

0 0 MHΔta2D 1

2aa1

⎞⎟⎟⎠⎛⎜⎜⎝

uH

uD

uV

ηS

⎞⎟⎟⎠ = 0,

where D =(d2,1 d2,2 d2,3 d2,4 d2,5 d2,6

), with

d2,1 = i 2h sin kh2 , d2,2 = i 2h cos (k−l)h

2 sin lh2 , d2,3 = i 2h cos lh

2 sin kh2 ,

d2,4 = i 2h cos kh2 sin lh

2 , d2,5 = i 2h cos (k−l)h2 sin kh

2 , d2,6 = i 2h sin lh2 .

Vanishing the 7 × 7 determinant of the matrix in the LHS of (4.9) again leads to apolynomial in E of degree nr, shown in Table 4.1.

4.4. The RT0−P0 pair. Equations (3.11) and (3.12) are employed to computethe discrete dispersion relation and the spatially discrete operators are obtained fromthe stencils in [27, Figures 3.1 and 3.2]. The velocity and elevation node locations areshown in Figure 4.2(c) with the chosen normal directions given in Figure 3.1.

Equation (3.12) is first discretized at the two typical surface-elevation nodes cor-responding to lower left and upper right triangles and identified with triangles K1 andK2, respectively, in Figure 4.2(c). Note that the piecewise-constant basis function ξKj

(j = 1, 2, 3, . . . ) is associated with node Kj , and it takes the value 1 on Kj and 0elsewhere. Further, we also identify ηh over triangle Kj with the piecewise-constantnodal value ηKj

, j = 1, 2, 3, . . . .

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At nodes K1 and K2, for ξh = ξK1and ξK2

, respectively, (3.12) leads to

a1 ηK1− 2HΔt

h2a2(J9 + J12 − J13) = 0,(4.10)

a1 ηK2− 2HΔt

h2a2(J13 − J14 − J16) = 0.(4.11)

We now discretize (3.11) at the three typical velocity nodes corresponding to the threepossible types of faces, i.e., horizontal, diagonal, and vertical. The discretization isperformed at nodes 5, 6, and 9 in Figure 4.2(c), and the support of the velocity basisfunctions at these nodes involves triangles K1, K6, K7, and K8. We obtain

a3(−J2 + 4 J5 − J8)− fΔt a2 (J2 + J4 + J6 + J8)− 6gΔt a2 (ηK6− ηK7

) = 0,(4.12)

2 a3 J6 − fΔt a2 (J2 − J5 − J7 + J9)− 6gΔt a2 (ηK7− ηK8

) = 0,(4.13)

a3(J7 − 4 J9 + J12)− fΔt a2 (J6 + J7 + J12 + J13)− 6gΔt a2 (ηK1− ηK8

) = 0.(4.14)

Again, the discrete velocity and surface-elevation nodal values are searched by con-sidering the behavior of one Fourier mode. Note that, as previously mentioned, weneed to consider three possible types of faces, i.e., horizontal, diagonal, and vertical(written as H , D, and V , respectively) and two typical barycenters correspondingto lower left and upper right triangles (C1 and C2, respectively). This leads us to

distinguish three amplitudes for the normal velocity, denoted by JH , JD, and JV , andtwo amplitudes named ηC1

and ηC2for the surface-elevation.

Substitution of the Fourier mode into the two discrete continuity equations (4.10)and (4.11), and the three discrete momentum equations (4.12)–(4.14), leads to thefollowing 5× 5 square matrix system for the Fourier amplitudes:

(4.15)

⎛⎝ N −gΔta2Dt

HΔta2D h2

2 a1I2

⎞⎠( J

η

)= 0,

where J = (JH , JD, JV ), η = (ηC1, ηC2

), I2 is the 2× 2 identity matrix, and

N =1

3

⎛⎜⎜⎝2a3 fΔta2b5 −(a3 − fΔta2)b6

−fΔta2b5 a3 fΔta2b4

−(a3 + fΔta2)b6 −fΔta2b4 2a3

⎞⎟⎟⎠ ,

with

D =

(−d3,1 d3,2 −d3,3d3,1 −d3,2 d3,3

),d3,1 = ei(k−2l) h

6 , d3,3 = ei(l−2k) h6 , b4 = cos kh

2 ,

d3,2 = ei(k+l) h6 , b6 = cos (k−l)h

2 , b5 = cos lh2 .

For the 5×5 determinant of the matrix system (4.15) to vanish we obtain a polynomialin E of degree nr, shown in Table 4.1.

4.5. The mass lumping technique. As mentioned in section 1, the use ofexplicit temporal schemes (the EF, AB2, and AB3 schemes in the present study)combined with mass lumping would allow us to compute the FE SW solution withoutsolving a linear system, and hence to combine the advantages of both fast and simple

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FD schemes and unstructured and flexible FE schemes. For the P0 − P1 and PNC1 −

P1 pairs, the velocity mass matrices are “naturally” diagonal. For the latter pairsuch a property results from the orthogonality of the nonconforming basis functions,as mentioned in section 3.2.2. Consequently, the lumping procedure need only beperformed for the P1 elevation mass matrix for these pairs. The lumping techniqueadopted in this study consists of adding to the diagonal elements of the consistentmass matrix the off-diagonal elements so that the total “mass” associated with a nodeis conserved [18]. If we let A and A be the consistent and lumped mass matrices,

respectively, we have Aij = (∑

k Aik) δij , where δij is the Kronecker delta. In practice,

the lumping of the P1 elevation mass matrix for the P0−P1 and PNC1 −P1 pairs leads

us to set a = 6 in (4.4) and (4.9), respectively. For the RT0 − P0 element, the masslumping technique for the velocity mass matrix requires that triangles K of Th satisfythe Delaunay property, as shown in [4]. The dispersion relation obtained in section 4.4for this pair needs thus to be recomputed on equilateral triangles, for example, whichis beyond the scope of this paper. Hence, mass lumping is considered only for theP0 − P1 and PNC

1 − P1 pairs in the following.

The behavior of the dispersion relations for the three FE pairs examined in thisstudy is analyzed in the next section. The analysis is performed without mass lumping,except when the lumping procedure is mentioned.

5. Analysis of the dispersion relations. The space and time discretizationof (2.1)–(2.2) should well approximate the continuous modes corresponding to thethree solutions of (2.4). In order to compare the discrete and the continuous solutions,(2.4) is rewritten as

(5.1) X3 + 2LX2 +(c2[(kh)2 + (lh)2] + F 2 + L2

)X + c2[(kh)2 + (lh)2]L = 0,

where X = iωΔt, F = fΔt, L = λΔt, and the wave Courant number (CFL param-eter) is c =

√gHΔt/h. The three solutions of (5.1) are denoted by Xj = iωAN

j Δt,

and we let EANj ≡ eXj = eiω

ANj Δt, j = 1, 2, 3.

We now analyze the discrete solutions Ej , j = 1, 2, 3, . . . , nr, of the polynomials(dispersion relations) obtained from (4.4), (4.9), and (4.15) for the three FE pairs andthe five time stepping schemes examined in section 4. Since we are mostly concernedwith stability issues, the preceding Fourier analysis is convenient as the amplificationfactor and phase errors per time step may be obtained from the dispersion relations.The main difficulty is to solve polynomials of order up to 21 with coefficients dependingon several parameters. Another approach (not examined in this study) is the use ofpropagation factors which give the amplitude and phase errors per wavelength [22].

In sections 5.1, 5.2, and 5.3, the gravity wave limit is first considered, where f = 0(in the absence of rotation), and we then comment on the more general case f �= 0 insection 5.4.

5.1. The discrete solutions. For the temporal schemes examined in this study,a few discrete solutions Ej , j = 1, 2, 3, . . . , nr, are given analytically in Table 5.1 for

the P0 − P1 pair with (s, r) = (1, 3) and the PNC1 − P1 pair with (s, r) = (2, 5), and

in Table 5.2 for the RT0 − P0 scheme, where

p1 = 3c2∑2

k=1 |d1,k|2, p1,s = a(aL2 − 8ps), p3,s =√p1,s(4L− 9L2 − 8 + 36

a ps),

p2 = c2∑6

k=1 |d2,k|2, p2,s =1a (9L− 4) p1,s, p4,s =

1a√p1,s

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for s = 1, 2, and

q1 = 12− 23L, q3 = q31 + 216L(3− 92L),

q2 = q21 + 242L, q4 = (q3 +√q23 − q32)

13 .

Complex numbers are expected for q4,√p1,s, p3,s and p4,s, s = 1, 2, depending on

L, kh, and lh. Note that all Ej , j = 1, 2, 3, . . . , nr, may not be obtained analytically,in which case they are computed under the Maple environment. This is the casefor the solutions E3r+1,...,3r+6 in Table 5.1 corresponding to the FE discretizations

P0 − P1 and PNC1 − P1 using the AB3 scheme, and for most of the results obtained

with the RT0 − P0 pair in Table 5.2.

Table 5.1

Solutions Ej, j = 1, 2, 3, . . . , for the P0 − P1 and PNC1 − P1 pairs and the EB, EF, CN,

AB2, and AB3 schemes. For the P0 − P1 pair we have (s, r) = (1, 3) and for the PNC1 − P1 pair

(s, r) = (2, 5). The solutions E3r+1,...,3r+6 for the AB3 scheme are obtained using Maple.

EB EF CN

E1,...,r = 11+L

E1,...,r = 1− L E1,...,r = 2−L2+L

Er+1,r+2 =a(L+2)±√

p1,s2a(L+1)+4ps

Er+1,r+2 =a(2−L)±√

p1,s2a

Er+1,r+2 =2a−ps±

√p1,s

a(L+2)+ps

AB2 AB3

E1,...,r = 14(2−3L+

√(3L−2)2+8L) E1,...,r = 1

36(q1 + q4 +

q2q4)

Er+1,...,2r = 14(2−3L−

√(3L−2)2+8L) Er+1,...,2r = 1

36(q1 + e

2iπ3 q4 + e

4iπ3

q2q4)

E2r+1,2r+2 = 12+

3(p4,s−L)

8± i

√p3,s

+p2,s

32ap4,s

E2r+1,...,3r = 136

(q1 + e4iπ3 q4 + e

2iπ3

q2q4)

E2r+3,2r+4 = 12− 3(p4,s+L)

8± i

√p3,s

−p2,s

32ap4,s

E3r+1,...,3r+6 (Maple)

Table 5.2

Solutions Ej , j = 1, 2, 3, . . . , for the RT0−P0 pair. When available, E1,2,3 are given analyticallyfor the EB, EF, CN, AB2 and AB3 schemes. The other solutions are obtained using Maple.

EB EF CN

E1 = 11+L

E1 = 1− L E1 = 2−L2+L

E2,...,5 (Maple) E2,...,5 (Maple) E2,...,5 (Maple)

AB2 AB3

E1 = 14(2−3L+

√(3L−2)2+8L) E1 = 1

36(q1 + q4 +

q2q4)

E2 = 14(2−3L−

√(3L−2)2+8L) E2 = 1

36(q1 + e

2iπ3 q4 + e

4iπ3

q2q4)

E3,...,10 (Maple) E3 = 136

(q1 + e4iπ3 q4 + e

2iπ3

q2q4)

E4,...,15 (Maple)

As observed in Tables 4.1, 5.1, and 5.2, the number of discrete solutions Ej ,j = 1, 2, 3, . . . , nr, is ranging from 5 to 21 for the FE pairs and temporal schemesexamined in this study, and thus it largely exceeds the number of continuous ones,i.e., the three solutions of (5.1). The number of spurious discrete solutions having nophysical basis is hence ranging from 2 to 18 depending on the discrete schemes.

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When f = 0, the three continuous solutions of (5.1) greatly simplify, and weobtain

(5.2) EAN1 = e−L, EAN

2,3 = e−L2 ± 1

2 i√

4c2[(kh)2+(lh)2]−L2.

The discrete solutions Ej , j = 1, 2, 3, . . . , nr, in Tables 5.1 and 5.2 are renamed:

ED1j for the P0 − P1 pair, ED2

j for the PNC1 − P1 pair, and ED3

j for the RT0 − P0

scheme. Further, such solutions of the discrete dispersion relations are divided intotwo groups depending on whether (section 5.3) or not (section 5.2) they are writtenin terms of kh and lh. These two cases are now examined.

5.2. The solutions Ej independent of kh and lh. Such solutions are inde-pendent of the space discretization scheme, and they are graphed in Figure 5.1 forL in the interval [0, 0.1]. Indeed, for all the timescales of interest and relevance tooceanic, atmospheric, and groundwater flows, we should have L � 0.1 in practice[30].

(a) EB, EF and CN (b) AB2 (c) AB3

L0.05 0.1

0.9

1

EBCN, ANEF

0L

0 0.05 0.10

0.2

0.4

0.6

0.8

1.0

1, AN

2

L0 0.05 0.1

0

0.2

0.4

0.6

0.8

1.0

1, AN

2

3

Fig. 5.1. The solutions |Ej |, j = 1, 2, 3, . . . , independent of kh and lh, as a function of L:

(a) |ED11,2,3| = |ED2

1,...,5| = |ED31 | for the EB, CN, and EF schemes, from top to bottom, respec-

tively. The continuous solution EAN1 in (5.2), also independent of kh and lh, is denoted by AN;

(b) |ED11,2,3| = |ED2

1,...,5| = |ED31 | (curve 1) and |ED1

4,5,6| = |ED26,...,10| = |ED3

2 | (curve 2) for the AB2

scheme; (c) |ED11,2,3| = |ED2

1,...,5| = |ED31 | (curve 1), |ED1

4,5,6| = |ED26,...,10| = |ED3

2 | (curve 2), and

|ED17,8,9| = |ED2

11,...,15| = |ED33 | (curve 3) for the AB3 scheme.

The solutions |ED11,2,3|, |ED2

1,...,5|, and |ED31 | for the EB, EF, and CN schemes are

displayed in Figure 5.1(a). All the curves are monotonically decreasing from 1, andhence all the temporal discretizations lead to stable results for these roots. Note thatthe root |E | = (2 − L)/(2 + L) for the CN scheme is very close to the continuoussolution EAN

1 = e−L in (5.2) corresponding to the curve AN in Figure 5.1.As shown in Table 5.1, Ej , j = 1, . . . , r, suggest the presence of multiple roots.

However, in the case of the P0 − P1 pair and the EB scheme, ED11 corresponding

to EAN1 usually depends on h, but ED1

1 = 11+L when F = 0, while ED1

2,3 are the

conjugate spurious modes 1+L±iF(L+1)2+F 2 given in Table 5.6 which also reduced to 1

1+L for

F = 0. Hence, the solutions ED11 and ED1

2,3 coincide here because F = 0. The modes

ED12,3 take the form of propagating inertial oscillations and have no particular spatial

characteristics. They are a consequence of using two times more velocity nodes thansurface elevation nodes [27]. Similar results hold for the PNC

1 −P1 pair; however, theinertial oscillations are double roots because there are three times more velocity nodes

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than surface elevation nodes. Note that the RT0 − P0 pair is free of such spuriousinertial modes [27], and hence 1

1+L is the single root in Table 5.2. Similar results alsohold for the EF and CN schemes.

Spurious inertial oscillations are also present for the AB2 and AB3 schemes. In-deed, assuming E1 corresponds to EAN

1 in Table 5.1, then E2,...,r are the spurious

inertial modes for both the P0−P1 (r = 3) and PNC1 −P1 (r = 5) pairs. Again, these

modes are given in Table 5.6 for F �= 0.However, when a τ -level time stepping scheme is used, with τ > 2, as is the case

for the AB2 (τ = 3) and AB3 (τ = 4) schemes in this study, additional spuriousmodes, distinct from the spurious inertial oscillations, are present. Indeed, it appearsthat for a τ -level time discretization scheme there are precisely τ − 1 nonmultiplesolutions independent of kh and lh. For the three FE pairs, without taking accountof the multiplicity of the roots, we obtain one solution for the EB, EF, and CN schemes(τ = 2), two solutions for the AB2 scheme (τ = 3), and three solutions for the AB3scheme (τ = 4), independent of kh and lh. When τ > 2, these additional (spurious)roots, ED1

4,5,6, ED26,...,10, and E

D32 for the AB2 scheme and ED1

4,...,9, ED26,...,15, and E

D32,3 for

the AB3 scheme, do not correspond to the analytic ones.The solutions |ED1

1,...,6|, |ED21,...,10|, and |ED3

1,2 | for the AB2 scheme are displayed

in Figure 5.1(b). The curves 1, corresponding to the solutions ED11,2,3 = ED2

1,...,5 =

ED31 , and AN are in good agreement, and hence these solutions are expected to

correspond to the continuous one. However, curve 2 corresponds to the spurious roots|ED1

4,5,6| = |ED26,...,10| = |ED3

2 |. The solutions |ED11,...,9|, |ED2

1,...,15|, and |ED31,2,3| for the AB3

scheme are displayed in Figure 5.1(c). The curves 1, corresponding to the solutions|ED1

1,2,3| = |ED21,...,5| = |ED3

1 |, and AN are again in good agreement, and these rootsshould correspond to the continuous solution. However, curve 2 again correspondsto the spurious roots |ED1

4,5,6| = |ED26,...,10| = |ED3

2 |, while curve 3 corresponds to the

spurious solutions ED17,8,9 = ED2

11,...,15 = ED33 .

Finally, for all the timescales of interest and relevance to oceanic, atmospheric,and groundwater flows, we should have L � 0.1 in practice, and hence all the rootsindependent of kh and lh should be stable, with |Ej | ≤ 1, j = 1, 2, 3 . . . . Further, the

roots Ej , j = 1, 2, 3 . . . , corresponding to the EB, EF, CN, AB2 schemes and ED14,5,6,

ED26,...,10, and E

D32 for the AB3 scheme are real whatever the choice of L. The other

roots for the AB3 scheme are complex numbers if and only if L ≥ 0.54897, which isunrealistic. Because in practice we have L� 0.1, all the solutions independent of khand lh, in addition to being stable, will be real with no phase error and no dispersion.

5.3. The solutions Ej dependent on kh and lh. We first comment on theimplicit temporal schemes and then examine the explicit ones. For each schemethe stability conditions and eventual damping are evaluated, but the phase error iscomputed only when the scheme may be of practical interest.

5.3.1. The implicit EB and CN schemes. For the EB and CN schemes, thesolutions ED1

4,5 , ED26,7 , and ED3

2,3 are considered as they correspond to the continuous

roots EAN2,3 in (5.2). Note that the complex conjugate solutions ED3

4,5 are spurious.The EB scheme is stable but at the expense of a severe damping. To estimate

the damping factor induced by the EB scheme, the quantity σ1 corresponding tomax(|ED1

4 |, |ED15 |), max(|ED2

6 |, |ED27 |), and max(|ED3

2 |, |ED33 |), is graphed in Fig-

ure 5.2. Such solutions reach their minimum and maximum values along the axiskh = lh independently of c, as observed using Maple, and thus σ1 is displayed interms of kh = lh and L. The wave Courant number is chosen to be c = 1, as we use

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P0 − P1 PNC1 − P1 RT0 − P0

EB σ1

00

L

0.3

kh0.1 π

1

00

L

0.3

kh0.1 π

1

000.3

Lkh

0.1 π

1

CN

σ2

0.11

Lkh

1.04

0 0

π0.11

Lkh

0

π

1.04

0

0.11

Lkh

0

1.04

0

π

θ

0-1.3

kh

0

0.1 π

0

L

1

0-1.2

kh

0

0.1 π

0

L

1

0-1.1

kh

0

0.1 π

0

L

1

Fig. 5.2. The values of σ1 are displayed in the case of the EB scheme and σ2 and θ are graphedin the case of the CN scheme in terms of kh = lh and L for c = 1.

an implicit scheme, and kh = lh and L vary over [0, π] and [0, 0.1], respectively. Thedamping is significant for the three FE pairs as kh increases from 0 to π, and it isweakly influenced by the bottom friction parameter L when c = 1.

We have |ED34 | = |ED3

5 |, as the two spurious roots are complex conjugate, with0.164 ≤ |ED3

4 | ≤ 0.229 for 0 ≤ kh, lh ≤ π, and 0 ≤ L ≤ 0.1. Again, the value ofL seems to have no influence on the results. Hence, these spurious modes do notcontribute significantly to the solution.

For the CN scheme, the solutions |ED15 |, |ED2

7 |, and |ED32 | correspond to |EAN

2 |,while |ED1

4 |, |ED26 |, and |ED3

3 | correspond to |EAN3 |. We denote by σ2 the quantities

|ED14 |/|EAN

3 |, |ED26 |/|EAN

3 |, and |ED32 |/|EAN

2 | and by σ3 the quantities |ED15 |/|EAN

2 |,|ED2

7 |/|EAN2 |, and |ED3

3 |/|EAN3 |. Again, since σ2 and σ3 reach their minimum and

maximum values along the axis kh = lh, they are graphed in terms of kh = lh and Lfor c = 1. The graphs of σ2 and σ3 coincide for the three FE pairs, and hence only σ2is displayed in Figure 5.2. Good agreement is obtained between the continuous andcomputed results for small values of L and for small wave numbers kh = lh, since σ2is close to 1. However, as L and kh = lh increase, σ2 departs slightly from 1 (within afew percent) but |ED1

4,5 |, |ED26,7 |, and |ED3

2,3 | remain smaller than 1. For 0 ≤ kh, lh ≤ π,

the spurious solutions |ED34 | = |ED3

5 | are close to 1 with |ED34 | = 1 for L = 0 and

0.9909 ≤ |ED34 | ≤ 0.9950 for L = 0.1.

The relative phase error, defined as θ = argEDC/ argEAN , is also computed forthe CN scheme, where EDC and EAN are the discrete FE solutions and the corre-

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sponding continuous ones, respectively. Ideally, we expect θ = 1. Again, the quanti-ties θ = argEDC/ argEAN reach their minimum and maximum values along the axiskh = lh, and they are thus graphed in terms of kh = lh and L for c = 1. Further, thegraphs for argED1

4 / argEAN3 and argED2

6 / argEAN3 exactly coincide with the graphs

for argED15 / argEAN

2 and argED27 / argEAN

2 , respectively, and argED32 / argEAN

2 co-incides with argED3

3 / argEAN3 . The quantity θ is displayed in Figure 5.2, and we ob-

serve that the phase is badly approximated by the three FE pairs for kh = lh ≥ π/2,i.e., for small wavelengths. Further, θ is weakly influenced by L when c = 1.

Finally, the results obtained for σ1, σ2, and θ when c = 0.5 are close to thosedisplayed in Figure 5.2 when c = 1.

5.3.2. The explicit EF scheme. The solutions ED14,5 , E

D26,7 , and ED3

2,3,4,5 are

now considered. Note that ED32,3 correspond to the continuous solutions EAN

2,3 in (5.2),

while the roots ED34,5 are spurious. Such spurious solutions may lead to instabilities,

contrary to the EB and CN schemes, and they are thus included in the present stabilityanalysis.

For L = 0, we obtain |E| > 1 for the three pairs while |EAN2 | = |EAN

3 | = 1, andhence the EF scheme is unconditionally unstable for these pairs. For example, in thedomain 0 ≤ kh, lh ≤ π with c = 0.05, |Er+1| = |Er+2| increases monotonically for thetwo pairs of Table 5.1 from 0 at the origin (kh, lh) = (0, 0) to 1.0296 for the P0 − P1

pair (with r = 3) and 1.0198 for the PNC1 − P1 scheme (with r = 5) at the point

(kh, lh) = (π, π). Similar results are obtained using the mass lumping technique,except that max(|ED1

4,5 |) = 1.0099 and max(|ED26,7 |) = 1.0066 at (kh, lh) = (π, π). For

the RT0−P0 pair, the root |ED32 | = |ED3

3 | (corresponding to the continuous solution)increases monotonically from 0 at the origin (kh, lh) = (0, 0) to 1.0149 at the point(kh, lh) = (π, π), while the spurious solution |ED3

4 | = |ED35 | has its maximum value

(1.0440) at the origin.For L �= 0, analytical computations show that sufficient stability conditions are

obtained for the P0 − P1 and PNC1 − P1 pairs in the case L ≥ 24c2 and L ≥ 16c2,

respectively. For the RT0 −P0 pair, the stability condition L ≥ 36c2 is obtained withMaple. The stability conditions for the three pairs are graphed as a function of thewave Courant number c in Figure 5.3(a) and they correspond to the area above thecurves 1, 2, and 3 for the PNC

1 − P1, P0 − P1, and RT0 − P0 pairs, respectively. Notethat for the RT0 − P0 pair, the spurious solutions ED3

4,5 are the most constraining in

terms of stability compared to the roots ED32,3 (as observed above in the case L = 0).

In order to determine when the stability conditions are physically relevant, twocases corresponding to river and deep ocean modeling are examined. For river mod-eling we may rewrite λu in (2.1) as λu = Cd| u |u/H , where Cd is a bottom dragcoefficient, to obtain the quadratic friction approximation. For test problems, Cd iscommonly on the order of 2.5 × 10−3, H = 10 m, and | u | = 0.4m s−1, which leadsto λ = 10−4 s−1. In deep ocean modeling, the term λu in (2.1) cannot be physicallylinked to a bottom drag coefficient as the bottom and surface currents may be verydifferent. In this case, λu is considered as a dissipative term, and the value of λ ischosen in order to match the mean flow with the observations. In the North Atlanticocean, for example, λ is such that λ ≤ 10−6 s−1.

To this regard, the results of Figure 5.3(a) show that the EF scheme imposessevere constraints on c in practice. Indeed, for c = 0.01 we need to choose L ≥0.0016 for the PNC

1 − P1 scheme, which has the less restrictive stability conditionsin Figure 5.3(a), to be stable. In river modeling, using H = h = 10 m leads toλ = L/Δt =

√gHL/ch ≥ 0.16 s−1, while in the deep ocean with H = 1000 m and

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(a) NML (b) ML

0.06

0.2

0.04

0.15

0.1

0.02

0.05

00

c0.1

L

1

2

3

0.08 0.06

L

1

2

0.08

0.04

0.06

0.04

0.02

0.02

00

c0.10.08

Fig. 5.3. The stability conditions for the EF time stepping scheme correspond to the areaabove the curves 1, 2, and 3 for the PNC

1 − P1, P0 − P1, and RT0 − P0 pairs, respectively. Thestability conditions are obtained (a) without (NML) and (b) with (ML) the use of the mass lumpingtechnique. Note that the lumping is not performed for the RT0 − P0 pair in this study.

h = 10 km we obtain λ ≥ 1.6 × 10−3 s−1. In both cases λ is too large to match thephysical requirements, and the EF scheme would cause excessive damping.

When mass lumping is performed, analytical computations show that sufficientstability conditions are obtained for the P0 − P1 and PNC

1 − P1 pairs in the casesL ≥ 8c2 and L ≥ 16

3 c2, respectively. Such conditions are graphed in Figure 5.3(b),

and they are three times less restrictive than the stability conditions obtained inFigure 5.3(a) without mass lumping. However, the EF scheme is still very constrainingfor c in practice as the upper bounds obtained above (without mass lumping) for λ inriver and deep ocean modeling are only divided by 3 when mass lumping is employed.

5.3.3. The explicit AB2 scheme. The solutions ED17,...,10, E

D211,...,14, and E

D33,...,10

are considered. The roots ED17,8 , E

D211,12, and E

D33,4 correspond to the continuous solu-

tions EAN2,3 , while the roots ED1

9,10, ED213,14, and E

D35,...,10 are spurious. Again, the spu-

rious solutions are taken into account in the following analysis as they may lead toinstabilities.

For L = 0, we obtain |E| > 1 for the three pairs while |EAN2 | = |EAN

3 | = 1, andhence the AB2 scheme is unconditionally unstable for these pairs, as it is for the EFone. However, contrary to the EF scheme, the instability for the AB2 scheme is weakin the absence of friction. Indeed, in the domain 0 ≤ kh, lh ≤ π and for c = 0.05, theconjugate roots |ED1

7,8 |, |ED211,12|, and |ED3

3,4 |, corresponding to the continuous solutions,increase monotonically from 0 at the origin (kh, lh) = (0, 0) for the three pairs to1.00102, 1.00043, and 1.00024 at the point (kh, lh) = (π, π) for the P0−P1, P

NC1 −P1,

and RT0 − P0 pairs, respectively. When mass lumping is performed, the behaviorof |ED1

7,8 | and |ED211,12| is unchanged, except that the maximum values at the point

(kh, lh) = (π, π) are now 1.000104 and 1.000046 for the P0 − P1 and PNC1 − P1

pairs, respectively, i.e., ten times less compared to the case without lumping. Theinstability is thus much weaker for these pairs when mass lumping is performed, andthe P0−P1 and PNC

1 −P1 schemes might eventually be used for a short time to obtainacceptable solutions. For the RT0−P0 pair, the spurious conjugate roots |ED3

5,6 | reachthe maximum value 1.00244 at the origin (kh, lh) = (0, 0), and, as observed for theEF scheme, they are the most constraining root in terms of stability. The remainingroots are small in magnitude and will not contribute significantly to the computed

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(a) NML (b) ML

0.06

0.02

0.04

0.015

0.01

0.02

0.005

00

c0.1

L

1

2

3

0.08 0.06

0.004

0.04

0.003

0.002

0.02

0.001

00

c0.1

L

1

2

0.08

Fig. 5.4. The same as for Figure 5.3, but for the AB2 time stepping scheme.

solution.For L �= 0 the stability conditions for the three pairs are graphed as a function

of the wave Courant number c in Figures 5.4(a)–(b) without and with mass lumping.As for Figure 5.3, these conditions correspond to the area above the curves 1, 2, and3 for the PNC

1 −P1, P0−P1, and RT0−P0 pairs, respectively. Note that the solutionsED1

7,8 , ED211,12, and ED3

5,6 are the most constraining roots in terms of stability, as theyare for L = 0. The stability conditions are clearly much less restrictive for the AB2scheme than for the EF one. Further, the conditions graphed in Figure 5.4(b) withlumping are roughly 10 times less constraining than those obtained in Figure 5.4(a)without lumping, as far as the P0 −P1 and PNC

1 −P1 pairs are concerned. Note thatthe latter pair has the less restrictive stability conditions in Figure 5.4, as is true forthe EF scheme.

If c = 0.02, for example, we need to choose L ≥ 2.3× 10−6 for the PNC1 −P1 pair

to be stable when the mass lumping procedure is employed. In river modeling, usingH = h = 10 m leads to λ ≥ 1.15×10−4 s−1, while in the deep ocean with H = 1000 mand h = 10 km we obtain λ ≥ 1.15 × 10−6 s−1. Such bounds give acceptable valuesand match the physical requirements. However, using c = 0.04 leads to λ ≥ 10−3 s−1

in river modeling with H = h = 10 m and to λ ≥ 10−5 s−1 in the deep ocean forH = 1000 m and h = 10 km. These results show that small wave Courant numbersneed to be employed in order to obtain physical lower bounds for L. Consequently,small values of c may prevent the efficiency of the numerical scheme.

5.3.4. The explicit AB3 scheme. The solutions ED110,15, E

D216,21, and ED3

4,...,15

are finally considered. The roots ED110,11, E

D216,17, and E

D34,5 correspond to the continu-

ous solutions EAN2,3 in (5.2), while the other roots are spurious. Again, the spurious

solutions are considered in the following analysis as they may lead to instabilities.(a) The case L = 0. After long and tedious algebra, the solutions ED1

10,15 and

ED216,21 are obtained as the roots of the polynomials

(5.3) E2(E − 1)± i

√ps72a

(23E2 − 16E + 5

)= 0,

with s = 1 for the P0 − P1 pair and s = 2 for the PNC1 − P1 pair. However, we were

unable to obtain such a simple polynomial form in the case of the RT0 − P0 pair.The AB3 scheme is conditionally stable for the three FE pairs examined in this

study. The maximum wave Courant numbers (c) that permit stability are computed in

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order to satisfy max(|ED110,15|, |ED2

16,21|, |ED34,...,15|) ≤ 1. Table 5.3 (middle column) shows

these maximum bounds of c, first without mass lumping (NML). The PNC1 − P1 and

RT0 − P0 pairs are the best and worst choice for stability, respectively.

Table 5.3

Maximum wave Courant numbers that ensure stability for the AB3 scheme with L = 0 and theP0 − P1, P

NC1 − P1, and RT0 − P0 pairs, without (NML) and with (ML) mass lumping.

NML ML

P0 − P1 0.1477 0.2558

PNC1 − P1 0.1809 0.3133

RT0 − P0 0.1206 —

The roots ED110,15, E

D216,21, and E

D34,...,15 are complex conjugate, and their absolute

values are graphed in Figure 5.5 for the P0 − P1, PNC1 − P1, and RT0 − P0 pairs

with c = 0.1477, 0.1809, and 0.1206, respectively, corresponding to upper bounds forstability (see Table 5.3). These discrete solutions are compared with the continuousones EAN

2,3 , and from (5.2) we obtain |EAN2 | = |EAN

3 | = 1 when L = 0. Except for

small wavelengths, |ED110 | = |ED1

11 | and |ED216 | = |ED2

17 | well agree with |EAN2 |, while

|ED34 | = |ED3

5 | mostly coincide with |EAN2 | over the whole spectrum with an error of

only 1% for small wavelengths. However, recall that among the three FE schemes,the RT0 − P0 pair has the smaller bound for c in Table 5.3 (c = 0.1206). Such asmall value of c is the most favorable case for the discrete solution to match withthe continuous one. Indeed, for c = 0.1206, much better results are obtained for theroots ED1

10,11 and ED216,17 compared to those graphed in Figure 5.5 with c = 0.1477 for

the P0 − P1 pair and c = 0.1807 for the PNC1 − P1 pair. Not only are the graphs of

|ED110 | = |ED1

11 | and |ED216 | = |ED2

17 | (not shown) still monotonic and take the value 1at the origin (kh, lh) = (0, 0) when c = 0.1206, but the minimum values at the point(kh, lh) = (π, π) are now 0.9530 and 0.9804 for the P0 − P1 and PNC

1 − P1 pairs,respectively, instead of being 0.8728 for both pairs with the values of c employed inFigure 5.5.

The remaining roots are spurious. Note that Maple has apparently reversed thegraphs of |ED3

6 | = |ED37 | and |ED3

10 | = |ED311 | in the neighborhood of kh = 0 and lh = 0,

leading to a discontinuous representation of these two solutions. The conjugate rootsED1

12,13, ED218,19, and ED3

10,11 first lead to unstable results for increasing values of c.The maximum values of c in Table 5.3 are in fact obtained in order to guaranteemax(|ED1

12,13|, |ED218,19|, |ED3

10,11|) = 1. The solutions Ej , j = 1, 2, 3, . . . , independent ofkh and lh and shown in Figure 5.1(c), have also been inserted into Figure 5.5 (secondcolumn) in the case L = 0 to give the full set of roots (j = 1, 2, . . . , nr) for the AB3scheme. These solutions are compared with their analytical counterpart EAN

1 = e−L

in (5.2). For L = 0, we obtain EAN1 = 1 and ED1

1,2,3 = ED21,...,5 = 1 and ED3

1 = 1 (sinceq4 = q2/q4 = q1 = 12 in Table 5.1); these values thus coincide with the continuoussolution. The other roots independent of kh and lh, namely, ED1

4,...,9, ED26,...,15, and

ED32,3 , are zero, and they correspond to spurious solutions.

The maximum wave Courant numbers (c) that permit stability are also written inTable 5.3 (right column) using the mass lumping procedure (ML) for the P0−P1 andPNC1 − P1 pairs. Compared to the case with no lumping (NML), the increase in c is

significant and about 73% for both pairs. Recall that mass lumping is not performedfor the RT0 − P0 scheme in this study. The solutions |Ej |, j = 1, 2, 3, . . . , nr, for theAB3 scheme with L = 0 and mass lumping are graphed in Figure 5.6 for the P0 − P1

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TEMPORAL SCHEMES FOR POINCARE WAVES IN FE SW MODELS 2237P0−P1

(c=

0.1477)

ED

11,2,3

=1,

ED

14,...,9=

0 |ED110 | = |ED1

11 | |ED112 | = |ED1

13 | |ED114 | = |ED1

15 |

π

0.9

lh0 πkh

0

1

π0

lhkh

0

0.5

0 π

1

π0

0.1

lhkh

0

0.2

0

0.3

π

PN

C1

−P1

(c=

0.1809)

ED

21,...,5

=1,

ED

26,...,15=

0 |ED216 | = |ED2

17 | |ED218 | = |ED2

19 | |ED220 | = |ED2

21 |

π

0.9

lhkh

0

0 π

1

π0

lhkh

0

0.5

0 π

1

π0

0.1

lhkh

0

0.2

0

0.3

π

RT0−P0

(c=

0.1206)

ED3

1=

1,E

D3

2,3

=0

|ED34 | = |ED3

5 | |ED36 | = |ED3

7 | |ED38 | = |ED3

9 |

00.99

ππkh

lh0

1

khlh

00π

0.4

π

0.8

0

00π

0.3

khlh

π0

0.6

|ED310 | = |ED3

11 | |ED312 | = |ED3

13 | |ED314 | = |ED3

15 |

khlh

0π

0.5

π0

1

00.88

π

0.92

π

0.96

khlh

0

0π

0.32

π

0.34

khlh

0

Fig. 5.5. The solutions |Ej|, j = 1, 2, 3, . . . , nr, for the AB3 scheme with L = 0. The results

for the P0 − P1, PNC1 − P1, and RT0 − P0 pairs are graphed for c = 0.1477, 0.1809, and 0.1206,

respectively, i.e., the maximum wave Courant numbers that ensure stability for these pairs.

and PNC1 −P1 pairs with c = 0.1477 and c = 0.1809, respectively. These values of c are

identical to those employed in Figure 5.5 (without lumping). Obviously, the solutionsindependent of kh and lh are unchanged. The roots |ED1

10 | = |ED111 | and |ED2

16 | = |ED217 |

now mostly coincide with |EAN2 | = |EAN

3 | = 1 over the whole spectrum with an errorof only 1% for small wavelengths, as is true for the RT0−P0 pair (without lumping).

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2238 DANIEL Y. LE ROUX, MICHEL DIEME, AND ABDOU SENEP0−P1

(c=

0.1477)

ED

11,...,6

=0,

ED

17,8,9

=1 |ED1

10 | = |ED111 | |ED1

12 | = |ED113 | |ED1

14 | = |ED115 |

π0.99

lh0 πkh

0

1

π0

kh

0

lh0

0.5

ππ

0

kh

0

0.1

lh

0.2

0 π

0.3

PN

C1

−P1

(c=

0.1809)

ED

21,...,10=

0,

ED

211,...,15=

1 |ED216 | = |ED2

17 | |ED218 | = |ED2

19 | |ED220 | = |ED2

21 |

π0.99

kh

0

lh0 π

1

π0

kh

0

lh0

0.5

ππ

0

kh

0

0.1

lh

0.2

0 π

0.3

Fig. 5.6. The solutions |Ej |, j = 1, 2, 3, . . . , nr, for the AB3 scheme with L = 0 and mass

lumping. The results for the P0 − P1 and PNC1 − P1 pairs are graphed for c = 0.1477 and 0.1809,

respectively. These values of c are identical to those employed in Figure 5.5 (without lumping). Notethat the maximum wave Courant numbers that ensure stability for these pairs, with the use of masslumping, are shown in Table 5.3 (ML).

The graphs of |ED112 | = |ED1

13 | and |ED218 | = |ED2

19 | reach a maximum of 0.5928 for bothpairs at the point (kh, lh) = (π, π), instead of 1 without lumping. However, when themaximum wave Courant numbers permitting stability are used with lumping, namely,c = 0.2558 for the P0−P1 pair and c = 0.3133 for the PNC

1 −P1 one (see Table 5.3), westill have max(|ED1

12,13|) = max(|ED218,19|) = 1, and the solutions ED1

12,13 and ED218,19 first

lead to unstable results. Again, the maximum values of c in Table 5.3 with lumping(ML) are obtained in order to guarantee max(|ED1

12,13|) = max(|ED218,19|) = 1. The

remaining spurious roots ED114,15 and ED2

20,21 are essentially unchanged compared tothe results of Figure 5.5 (without lumping). Finally, the improvement observed withmass lumping for small wavelengths in the case of the AB3 scheme is also observedfor the EF and AB2 schemes (results not shown) as far as the physical solutions areconcerned.

As for the CN scheme, The relative phase error, defined as argEDC/ argEAN , isalso computed for the AB3 scheme, where EDC and EAN are the discrete FE solutionsand the corresponding continuous solutions, respectively. Ideally, we expect θ =1. The quantities argED1

10 / argEAN1 = argED1

11 / argEAN2 and argED2

16 / argEAN1 =

argED217 / argE

AN2 are graphed in Figure 5.7 for c = 0.1477 and 0.1809, respectively,

without (NML) and with (ML) mass lumping. The results are considerably improvedcompared to those of Figure 5.2 (last line) for the CN scheme. Unfortunately, wewere unable to obtain coherent results for the RT0−P0 pair in the NML case, due totruncation errors.

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P0 − P1 PNC1 − P1

NML

01

kh

π1.1

lh

1.2

π 0

0kh

π0.95

lh

1

π

1.05

0

ML

0

0.7

khlh

0

π π

1

0

0.7

khlh

0

π π

1

Fig. 5.7. The relative phase errors argED110 / argEAN

1 and argED216 / argEAN

1 are graphed forthe AB3 scheme and L = 0, without (NML) and with (ML) mass lumping, in the case of the P0−P1and PNC

1 − P1 pairs with c = 0.1477 and 0.1809, respectively.

(b) The case L �= 0. The solutions ED110,...,15, E

D216,...,21 (shown in Figure 5.5 for

L = 0) are now obtained as the roots of the polynomials

(5.4) E2(E − 1) +1

24

(23E2 − 16E + 5

)(L± i

√8psa

− L2

)= 0,

after long and tedious algebra, with s = 1 for the P0 − P1 pair and s = 2 for thePNC1 − P1 pair. Note that (5.4) reduced to (5.3) when L = 0. Again, we were unable

to obtain such a simple polynomial form in the case of the RT0 − P0 pair.

Table 5.4

Increase (sign +) or decrease (sign −) in percent (%) between the solutions |ED111,12|, |ED2

17,18|,|ED3

5,10| for L �= 0 and the corresponding ones obtained in Figure 5.5 for L = 0, at particular points

(kh, lh). The P0 − P1, PNC1 − P1, and RT0 − P0 pairs are used with c = 0.1477, c = 0.1809, and

c = 0.1206, respectively, and two cases are considered: L = c 10−4 and L = c 10−3.

L = c 10−4 L = c 10−3

P0 − P1 |ED111 |, kh = lh = 0 − 1.5× 10−3 − 1.5× 10−2

c = 0.1477 |ED112 |, kh = lh = π +7.3× 10−4 +7.3× 10−3

PNC1 − P1 |ED2

17 |, kh = lh = 0 − 1.8× 10−3 − 1.8× 10−2

c = 0.1809 |ED218 |, kh = lh = π +8.9× 10−4 +8.9× 10−3

RT0 − P0 |ED35 |, kh = lh = 0 − 1.2× 10−3 − 1.2× 10−2

c = 0.1206 |ED310 |, kh = lh = 0 +5.9× 10−4 +5.9× 10−3

The roots ED110,...,15, E

D216,...,21, and ED3

5,...,15 shown in Figure 5.5 for L = 0 areno longer complex conjugate when L �= 0. Further, they are slightly influenced byL for realistic values of the bottom friction coefficient. Let us recall that we haveL = cλh/

√gH, and for both river and ocean modeling parameters used above we

obtained L = c 10−4.

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As in Figure 5.5, we still have |ED110 | = |ED2

16 | = |ED34 | = 1 at the point (kh, lh) =

(0, 0) for c = 0.1477, c = 0.1809, and c = 0.1206, respectively, when L �= 0, butnow |ED1

11 | < 1, |ED217 | < 1, and |ED3

5 | < 1. This expected damping of the discreteroots (corresponding to the continuous solution) is given in Table 5.4 (sign −) forL = c 10−4 and L = c 10−3.

More surprisingly, the roots ED112,13, E

D218,19, and E

D310,11, which first lead to unsta-

ble results at the point (kh, lh) = (π, π) in Figure 5.5 when L = 0, are such that|ED1

12,13| > 1, |ED218,19| > 1, and |ED3

10,11| > 1 for c = 0.1477, c = 0.1809, and c = 0.1206,respectively, at (kh, lh) = (π, π) when L �= 0. Such an increase in magnitude forthese spurious roots is shown in Table 5.4 (sign +) for L = c 10−4 and L = c 10−3. Itsuggests that slightly smaller values of c might be needed for the P0 −P1, P

NC1 −P1,

and RT0 − P0 pairs when L �= 0 compared to the case L = 0 in order to keep thesolution stable, particularly for long-term simulations.

Finally, the case L �= 0 has a negligible impact on the other solutions examinedin Figure 5.5 for L = 0, namely, the roots ED1

14,15, ED220,21, E

D36,...,9, and E

D312,...,15.

5.3.5. Summarizing results. The stability results of section 5.3 are summa-rized in Table 5.5 for the EB, CN, EF, AB2, and AB3 schemes.

Table 5.5

Summarizing stability results obtained in section 5.3 for the EB, CN, EF, AB2, and AB3schemes.

EB Stable but significant damping for c = 1.

CNStable with very small damping for c = 1; however, the phase is badlyapproximated for c = 1.

EF

L = 0: unconditionally unstable.

L �= 0: conditionally stable but not usable in practice because thestability conditions are not physically relevant.

AB2

L = 0: unconditionally unstable. However, the instability is weakwith mass lumping compared to the EF case.

L �= 0: conditionally stable, but very small values of c need to beemployed in order to obtain physical lower bounds for L.

AB3

L = 0: conditionally stable, the maximum values of c ensuring stabilityare given in Table 5.3. The phase is accurate.

L �= 0: conditionally stable, but smaller values of c might be neededcompared to the case L = 0 for long term simulations (Table 5.4).

5.4. The case f �= 0. Assume the Coriolis parameter is a constant differentfrom zero. For the temporal schemes employed in this study, only a few discretesolutions Ej , j = 1, 2, 3, . . . , nr, shown in Table 5.6, may be obtained analytically for

the P0 − P1 (r = 2) and PNC1 − P1 (r = 4) pairs, with

p =(12E2(E − 1) + (L+ iF )(23E2 − 16E + 5)

)×(12E2(E − 1) + (L − iF )(23E2 − 16E + 5)

),

q±5 = 4− 4(L± iF ) + 9(L± iF )2.

The remaining solutions, namely, Er+1,...,r+3 for the EB, EF, and CN schemes,E2r+1,...,2r+6 for the AB2 scheme and E3r+1,...,3r+9 for the AB3 scheme, and all thesolutions obtained for the RT0−P0 pair, are computed under the Maple environmentwhen possible.

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Table 5.6

Solutions Ej, j = 1, 2, 3, . . . , for the P0 − P1 (r = 2) and PNC1 − P1 (r = 4) pairs when f �= 0

in the case of the EB, EF, CN, AB2, and AB3 schemes. The solutions Er+1,...,r+3 for the EB,EF, and CN schemes, E2r+1,...,2r+6 for the AB2 scheme, and E3r+1,...,3r+9 for the AB3 schemeare obtained using Maple.

EB EF CN

E1,...,r = L+1±iF(L+1)2+F2 E1,...,r = 1− L± iF E1,...,r = −L2+(F±2i)2

(L+2)2+F2

Er+1,...,r+3 (Maple) Er+1,...,r+3 (Maple) Er+1,...,r+3 (Maple)

AB2 AB3

E1,...,r = 12− 3

4(L+ iF )± 1

4

√q+5 E1,...,3r : the 3r roots of pr/2

Er+1,...,2r = 12− 3

4(L− iF )± 1

4

√q−5 E3r+1,...,3r+9 (Maple)

E2r+1,...,2r+6 (Maple)

The Coriolis parameter varies very smoothly since we have f = 2Ω sinφ0, whereΩ = 7.29× 10−5 rad s−1 is the angular frequency of the earth’s rotation and φ0 is thelatitude. Since 0 ≤ φ0 ≤ π/2, we deduce that 0 ≤ f ≤ 1.458×10−4 s−1. Further, recallthat we have Δt = ch/

√gH, and for river (H = h = 10 m) and ocean (H = 1000 m,

h = 10 km) modeling parameters used above, we obtain F ≤ 1.458 × 10−4 c andF ≤ 1.458× 10−2 c, respectively.

The results of section 5 have been recomputed using F = 1.458× 10−2, i.e., themaximum bound above obtained for c = 1, in the case of the P0 − P1 and PNC

1 − P1

pairs. For Figures 5.1–5.4 and Table 5.3, the graphs and results are identical to thecase F = 0 within an error of the order of 1%, and the comments made in section 5are still valid. However, we were unable to graph the figures for the three FE pairsand the results of the whole section 5 for the RT0 −P0 pair in the case F �= 0, due tothe difficulty of solving high degree polynomial equations.

6. Numerical results. In order to validate the results obtained in section 5,we numerically simulate the propagation and dispersion of fast Poincare waves in acircular basin [23]. Among the temporal schemes previously examined, only the AB3scheme is considered in the following as it is of most practical interest, particularlywhen the mass lumping procedure is employed (see Table 5.5). Indeed, the EB schemeoverdamped the solution, while the EF and AB2 schemes may lead to stable resultsbut at the expense of employing very constraining (small) values of c that may preventthe efficiency of the numerical scheme. Further, since the results of section 5 are onlyslightly influenced by the bottom friction and Coriolis coefficients, for realistic valuesof these parameters, we set f = λ = 0 and hence F = L = 0 in what follows.

The circular domain has a radius of R = 1000km and is discretized using struc-tured and unstructured triangulations Th. The initial conditions are u(x, 0) = 0

and η(x, 0) = μe−ν(x2+y2). We let μ = 100m and H = 2000m, and hence theinitial perturbation amplitude represents 5% of the total depth. By choosing ν =6.4 × 10−11m−2, the e-folding radius of the initial Gaussian (the distance from theorigin for which η = μe−1) is 125 km.

The analytical solution of the problem is derived from (2.1)–(2.2) by exploitingthe circular symmetry since f = 0 and λ = 0. A second-order wave equation is thenobtained in polar coordinates for η, namely,

(6.1) ηtt − gH1

r(rηr)r = 0,

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subject to the boundary conditions ηr|r=R = 0 and no singularity at the origin, where√gH is the phase speed of the surface gravity waves. The initial conditions are

η(r, 0) = μe−νr2 and ηt(r, 0) = 0; the latter condition follows from setting the initialvelocity to zero. The exact solution of (6.1) is the Bessel-function expansion

η EX(r, t) =

∞∑n=1

κnJ0(√ρnr) cos gH

√ρnt,

where κn =∫R0

η(r,0)J0(√ρnr)r dr∫

R0

J20 (

√ρnr)r dr

and ρn are the roots of J1(√ρnR) = 0.

We first consider Mesh 1, made up of biased right isosceles triangles (as in Fig-ure 3.1) in order to validate the analytical results obtained on identical discretizations,and the uniform surface-elevation node spacing is h = 10 km. Since the domain iscircular, Mesh 1 crudely represents the domain boundary by a series of artificial steps,and hence the numerical experiments are stopped before the wave reaches the bound-ary.

(a) (b) (c)

Fig. 6.1. Numerically simulated surface elevation ηNS at different stages of the wave evolutionafter the wave has propagated over (a) 250 km, (b) 500 km, and (c) 750 km. The minimum andmaximum values of ηNS are approximately (a) −20, 20, (b) −10, 15, and (c) −7, 13.

The numerically simulated (NS) surface elevation ηNS is shown in Figure 6.1at different stages of gravity-wave propagation and dispersion after the wave haspropagated over (a) 250 km, (b) 500 km, and (c) 750 km, respectively. Using thePNC1 −P1, P0−P1, and RT0−P0 pairs without lumping, with appropriate values of Δt

required to ensure stability, leads to indistinguishable results (interpolated linearly).

Table 6.1

Minimum and maximum values of the numerically simulated (NS) and exact (EX) surfaceelevations in the case of the P0 − P1 scheme. For both the NML and ML cases, the minimum andmaximum values of ηNS and η EX are displayed at selected time steps (t.s.) in order to determinewhich values of Δt (in seconds) lead to stable (top) or unstable (bottom) results.

NML ML

(t.s.) ηNS ηEX (t.s.) ηNS ηEX

Δt=

11.3 (530) −6.483, 11.999 −6.533, 12.006

Δt=

19.4 (315) −6.514, 12.288 −6.447, 12.403

(540) −6.403, 12.328 −6.453, 12.353 (320) −6.448, 13.117 −6.380, 13.317

(550) −6.327, 13.434 −6.375, 13.419 (325) −6.383, 14.945 −6.315, 15.347

Δt=

11.4 (530) −6.453, 12.105 −6.495, 12.115

Δt=

19.5 (315) −6.492, 12.488 −6.425, 12.616

(540) −75.07, 74.16 −6.416, 12.729 (320) −6.938, 13.574 −6.359, 13.839

(550) −113.8, 116.4 −6.338, 14.480 (325) −15.17, 15.691 −6.292, 16.147

In order to determine upper bound stability limits for Δt, the minimum andmaximum values of the numerically simulated (NS) and exact (EX) surface elevations

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Table 6.2

The same for Table 6.1, but in the case of the PNC1 − P1 pair.

NML ML


Δt=

14.0 (450) −6.313, 15.063 −6.318, 15.223

Δt=

24.2 (255) −6.654, 12.649 −6.405, 12.872

(455) −6.268, 16.690 −6.270, 16.936 (260) −6.574, 14.470 −6.323, 15.025

(465) −6.181, 19.586 −6.160, 20.056 (265) −6.499, 17.161 −6.239, 17.952

Δt=

14.1 (450) −6.558, 16.117 −6.287, 16.330

Δt=

24.3 (255) −6.635, 12.894 −6.388, 13.163

(455) −11.17, 17.70 −6.237, 18.010 (260) −12.66, 15.06 −6.306, 15.668

(465) −31.28, 29.25 −6.114, 20.933 (265) −26.98, 25.02 −6.219, 18.557

Table 6.3

The same for Table 6.1, but in the case of the RT0 − P0 scheme without lumping (NML).

Δt = 8.7 Δt = 8.8


(610) −7.091, 12.433 −7.085, 12.441 (610) −7.054, 12.467 −7.030, 12.376

(620) −7.013, 12.338 −7.007, 12.349 (620) −8.482, 12.307 −6.953, 12.285

(650) −6.795, 12.078 −6.787, 12.090 (650) −18.89, 12.038 −6.735, 12.033

η are compared in Tables 6.1, 6.2, and 6.3 in the case of the P0 − P1, PNC1 − P1, and

RT0 −P0 pairs, respectively, for selected time steps (t.s.), with and without lumping.For example, the P0 − P1 scheme is unstable without lumping for Δt = 11.4 s, andhence the upper bound stability limit is assumed to be Δt = 11.3 s. Because g = 9.81ms−2, H = 2000 m, and h = 10 km, we deduce c =

√gHΔt/h = 0.1583. The

maximum wave Courant numbers c that ensure stability for the remaining pairs,both without and with lumping, are obtained in the same manner, and the resultsare summarized in bold in Table 6.4. Further, the maximum values of c, obtainedanalytically from the discrete schemes and shown in Table 5.3, are also reported forcomparison.

Table 6.4

Numerically simulated (NS) and discrete analytical (DC) results obtained for the maximumvalues of the wave Courant number that ensure stability. The P0 − P1, PNC

1 − P1, and RT0 − P0pairs are considered for both the NML and ML cases.

NML ML

NS DC NS DC

P0 − P1 0.1583 0.1477 0.2717 0.2558

PNC1 − P1 0.1961 0.1809 0.3390 0.3133

RT0 − P0 0.1219 0.1206 — —

The apparent discrepancies observed in Table 6.4 between the numerically simu-lated and discrete analytical results obtained for the maximum values of c ensuringstability range from 1% (for the RT0−P0 pair) to 8% (for the PNC

1 −P1 pair). Suchdifferences are mainly due to the fact that weak instabilities slowly increase with time.Hence, the upper bound stability limits obtained for Δt in Tables 6.1, 6.2, and 6.3should be smaller (from 1% to 8%), leading to smaller values of c (for the NS case in

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Table 6.4).Numerical simulations have also been performed on Mesh 2, an unstructured mesh

without smoothing with hmin = 5.263 km and hmax = 11.385 km, where hmin andhmax are the minimum and maximum edge lengths of Mesh 2, respectively. Althoughno discrete analytical results are available for comparison, the numerically simulatedresults obtained on Mesh 2 for the maximum values of c ensuring stability are closeto those reported in bold in Table 6.4 (the NS case on Mesh 1). However, c has to becomputed with values of h ranging from 5.286 km (for the RT0−P0 pair) to 5.929 km(for the P0 − P1 one), i.e., close to hmin.

Finally, the numerical results have been recomputed for f = 10−4 s−1, i.e., atmidlatitudes, but in such a case (f �= 0) the exact solution is not available. For thethree FE pairs, with and without mass lumping, the results are similar (within a fewpercent) to those obtained for f = 0 in Tables 6.1, 6.2, and 6.3 for ηNS , and Table 6.4is still valid.

7. Conclusion. This study examines the dispersion relation and spurious modebehavior for FE solutions of the 2D linearized SW equations based on the examinationof the P0 − P1, P

NC1 − P1, and RT0 − P0 FE pairs, and the backward and forward

Euler, Crank–Nicolson, and second and third order Adams–Bashforth time steppingschemes are employed. For each pair, the frequency wavenumber or dispersion relationis obtained and analyzed, and the stability properties are compared graphically withthe continuous case to illustrate the main points of interest. It is shown that for allpairs a number of solutions independent of the spatial discretization exist. However,in addition to being stable, these are shown to be real with no phase error and nodispersion for realistic parameters. For the other solutions (depending on the spatialdiscretization) the backward Euler and Crank–Nicolson schemes are stable, but theformer scheme induces a severe damping, while the phase is badly approximated withthe latter. The forward Euler and second order Adams–Bashforth schemes are shownto impose strong restrictive stability constraints on the wave Courant number, bothwith and without mass lumping. For the third order Adams–Bashforth scheme, amaximum bound for the wave Courant number is obtained for all pairs and has beenfound quite insensible to the variation of the physical parameters employed in thisstudy. The PNC

1 −P1 and RT0 −P0 pairs are the best and worst choices for stability,respectively. Finally, the use of mass lumping allows us to significantly increase themaximum wave Courant numbers permitting stability.

The preceding analysis illustrates how the Fourier approach can help in the se-lection of spatial and time discretization schemes. However, because of its restrictivenature, due to the use of constant values for the meshlength h and mean depth Hparameters, such an analysis should be only one step of the analysis process. The useof nonconstant h and H will be analyzed in subsequent studies.

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Documents

Time Discretization Schemes for Poincaré Waves in Finite-Element Shallow-Water Models