Implementation of a Discontinuous Galerkin Morphological Model Mirabito

8/10/2019 Implementation of a Discontinuous Galerkin Morphological Model Mirabito

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Implementation of a discontinuous Galerkin morphological modelon two-dimensional unstructured meshes

C. Mirabito a, , C. Dawson a, E.J. Kubatko b, J.J. Westerink c, S. Bunya da Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX 78712, USAb Department of Civil and Environmental Engineering and Geodetic Science, The Ohio State University, Columbus, OH 43210, USAc Environmental Fluid Dynamics Laboratories, Department of Civil Engineering and Geological Sciences, University of Notre Dame, Notre Dame, IN 46556, USAd Science and Safety Policy Research Division, Mitsubishi Research Institute, Inc., 2-3-6 O temachi, Chiyoda-ku, Tokyo 100-8141, Japan

a r t i c l e i n f o

Article history:Received 26 February 2010Received in revised form 10 August 2010Accepted 12 August 2010Available online 26 August 2010

Keywords:Shallow waterSediment transportBed morphologyFinite elementsDiscontinuous Galerkin

a b s t r a c t

The shallow water equations are used to model large-scale surface ow in the ocean, coastal rivers, estu-aries, salt marshes, bays, and channels. They can describe tidal ows as well as storm surges associatedwith extreme storm events, such as hurricanes. The resulting currents can transport bed load and sus-pended sediment and result in morphological changes to the seabed. Modeling these processes requirestightly coupling a bed morphology equation to the shallow water equations. Discontinuous Galerkinnite element methods are a natural choice for modeling this coupled system, given the need to solvethese problems on unstructured computational meshes, as well as the desire to implement hp-adaptivityfor capturing the dynamic features of the solution. However, because of the presence of non-conservativeproducts in the momentum equations, the standard DG method cannot be applied in a straightforwardmanner. To rectify this situation, we summarize and follow an extended approach described by Rheber-gen et al., whichuses theoretical results due to Dal Maso et al. appearing in earlier work. In this paper, wefocus on aspects of the implementation of the morphological model for bed evolution within theAdvanced Circulation (ADCIRC) modeling framework, as well as the verication of the RKDG method

in both h (mesh spacing) and p (polynomial order). This morphological model is applied to a numberof coastal engineering problems, and numerical results are presented, with attention paid to the effectsof h- and p-renement in these applications. In particular, it is observed that for sediment transport,piecewise constant (i.e., nite volume) approximations of the bed are very over-diffusiveand lead to poorsediment solutions.

2010 Elsevier B.V. All rights reserved.

1. Introduction

The modeling of ow and transport in coastal waters requires adetailed knowledge of winds, waves, currents, sediment transportand ultimately, the resulting morphological changes in the seabedthat occur as a result of these processes. Theerosion and depositionof bed sedimentcan have a majordetrimental impacton thecoastalpopulation, infrastructure, and environment. For example, the USArmy Corps of Engineers (USACE) maintains more than 12,000miles of waterways for transportation, which carry approximatelyone-sixth of the US inter-city freight [1]. Maintenance of thesewaterways through dredging and backlling operations representsa signicant cost to the USACE as well as other agencies and indus-tries.As another example, thestructural integrityof bridges, levees,

and piers can be compromised by excessive scour of the seabedaround thestructure.Also, dune, barrier island, andchannel erosionduring a hurricane leads to the removal of major ow controls,which signicantly affects inland inundation. In addition to theseinfrastructure-related issues, there exists a host of environmentalconcerns, such as beach erosion and the transport of contaminantswith sediment, which may actually act as a source or sink for con-taminants depending on the surrounding physico-chemical condi-tions [2].

Collectively, the various uid ow and transport processes thatlead to morphological changes in the seabed form an interdepen-dent physical system in which the uid motion, due to both wavesand currents, drives the transport of sediment, which dictates themorphological evolution of the seabed. In turn, the uid motion it-self is then directly affected by the morphological changes in thebed that it induces. Comprehensive modeling of these processesin the coastal zone presents several challenges andopen questions.Most existing hydrodynamic models use a xed-bed approach;that is, the bottom boundary of the seabed is not allowed to evolve

0045-7825/$ - see front matter 2010 Elsevier B.V. All rights reserved.doi: 10.1016/j.cma.2010.08.004

Corresponding author. Tel.: +1 512 471 1721.E-mail addresses: [email protected] (C. Mirabito), [email protected]

(C. Dawson), [email protected] (E.J. Kubatko), [email protected] (J.J. Westerink),[email protected] (S. Bunya).

Comput. Methods Appl. Mech. Engrg. 200 (2011) 189207

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in response to the uid motion. With respect to movable-bedmod-els, there is no single, generally accepted mathematical model thatis used, with descriptions of the uid-bed interface region varyingfrom simple single interface models to more complex three-phaseapproaches that consist of uid, uid-sediment, and sedimentphases; see, for example, [3]. Once an appropriate mathematicalmodel is formulated, there exists the numerical challenge of cou-pling models of various processes that exhibit disparate timescalesboth within the uid motion itself and between the uidand bed motion. The evolution of the bed is typically on a muchslower time scale compared to even the slowest time scale of theuid motion; a situation where this may not be the case is duringan extreme event, such as a hurricane.

In this paper, we focus on the coupling of shallow water hydro-dynamics with sediment transport described as bed load, and theapplication of an hp RungeKutta discontinuous Galerkin (RKDG)approximation to this coupled system. The mathematical modelwe consider consists of the depth-averaged (two-dimensional)shallow water equations and the well-known Exner equation [4] ,which describes the bed morphology. The models are coupled inthe sense that the sediment ux depends on the current, whilethe transport of sediment induces dynamic bathymetry, which af-fects the water depth. In addition, the fact that bathymetry is nowan unknown in the model rather than a known xed quantity givesrise to a non-conservative product in the shallow water momen-tum equations, which requires special treatment.

The RKDG method is an extension of nite volume methodswhich allows for arbitrary order of approximation in space and isapplicable to general unstructured meshes. A number of research-ers have studied nite volume methods for circulation/sedimenttransport/bed morphology models of various levels of complexity,in one and two space dimensions; see for example [3,512] . Thesepapers shed light on approaches for coupling the equations in bothspace and time, including the development of numerical uxesappropriate for DG discretizations.

Recently, Tassi et al. [13] have examined river bed evolution inshallow water ow using the RKDG method. The DG spatial dis-cretization that is used is an extended approach developed by Rhe-bergen et al. [14] to handle the non-conservative product that isintroduced when coupling the Exner equation to the shallow waterequations. We follow this approach here, but focus on the verica-tion of the RKDG method in both h (mesh spacing) and p (polyno-mial order), and on the application of the model to a number of coastal engineering problems, again emphasizing the effects of h-and p-renement in these examples.

Theworkdescribed hereinis thelatest ina seriesof papers bytheauthors and collaborators on the application of discontinuousGalerkin (DG) methods to shallow water ows [1522] . The RKDGmethod has been implemented within thegeneral AdvancedCircu-lation (ADCIRC) modeling framework, which is a parallel simulator

for shallow water ow and transport coupled with wind, wind-waves and other external forcings. The operational ADCIRC model,which is based on a continuous Galerkin method applied to theso-called Generalized Wave ContinuityEquation (GWCE), has beenusedto study a variety of coastalengineeringapplications, includingstorm surges generated by hurricanes and tropical storms [23,24] .The coupling of the GWCE-based formulation with a DG methodfor bed morphology has been studied by the authors in [25] ; how-ever, this coupling suffers from thefact that theGWCEmethod doesnot produce locally conservative currents, which can lead to spuri-oussedimenttransport solutions. One of theimmediategoals of ourresearch is the development of a DG-based hydrodynamic stormsurge model coupled to sediment transport, for modeling scourdue to strong currents during extreme storm events.

The remainder of this paper is outlined as follows. In Section 2we present the governing equations for hydrodynamics and bed

morphology. As mentioned above, the presence of dynamicbathymetry gives rise to a non-conservative product in the formu-lation. The analysis of such models is due to Dal Maso et al. [26] ,and we summarize their theoretical results. This formulation isthe basis of the RKDG method as originally presented by Rheber-gen et al. [14] . This formulation is described in Section 3 along withsome details of the implementation. In Section 4, several numericalstudies are presented. These studies begin by highlighting the con-vergence of the method in both h and p for a problem with an ana-lytical solution. We then focus on a number of coastal engineeringapplications, including ow through a converging channel, scour-ing around a bridge pier, and ebb and ood shoal formation in aninlet. In each application, we study the effects of h- and/or p-rene-ment within the RKDG formulation. We conclude in Section 4.5with some remarks on parallel performance.

2. Model description

2.1. Governing equations

The governing equations for hydrodynamics are derived byintegrating the NavierStokes equations with kinematic boundaryconditions over depth, assuming a hydrostatic pressure distribu-tion as well as a uniform velocity prole in the vertical, and ignor-ing the effects of diffusion. The following system of equations areobtained:

@ t H r q 0; 1a@ t q r Fmom gH r b smom ; 1bwhich, taken together, are known as the SaintVenant or shallowwater equations (SWE), which are dened on the domain X (0, T ]with appropriate initial and boundary conditions specied.

In (1a) and (1b) , H = f + b is the total depth of the seawater col-umn, where f is the observed free surface elevation measured po-sitive upward from a specied horizontal datum and b is the

bathymetric depth measured positive downward from the datum,as shown in Fig. 1. The vector q = (Hu ,Hv)T is the discharge rate,where u and v are the depth-averaged velocities in the x- and y-directions, respectively. The symmetric tensor

Fmom Hu2 12 gH

2 Huv

Huv H v 2 12 gH 2 !;

where the constant g = 9.80665 m/s 2 is standard gravity, and smomrepresents terms such as bottom friction, turbulent eddy viscosity,and additional surface forces from the Coriolis force and tide-gener-ating potential as well as surface stresses caused by wind, variableatmospheric pressure, andradiationstresses fromshort wind-drivenwaves.Note that H and q arebothfunctions of time t aswellas space.

Fig. 1. The seawater column, showing the relationship between H , f , and b.

190 C. Mirabito et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 189207


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In addition, sediment mass balance within the water column isconsidered. Under the assumptions of constant porosity e0 andignoring the effects of diffusion, the following sediment transportequation arises:

@ b@ t

11 e0 r q b 0; 2

where qb is the total sediment transport rate. This is an empirically

dened function of H , kuk2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 v 2p , and several other variables;see, for example, [27] . In the numerical results presented in this pa-per, only bed load transport due to currents is considered, and qb isa taken as a function of kuk

32 except where otherwise noted.

For the purposes of this paper, the coupled system consisting of Eqs. (1a), (1b), and (2) will be expressed as

@ t w r F Gr w s 0; 3where w = (H ,Hu ,Hv,b)T , F q j Fmom j qb

T ; Gr w 0; gH @ xb; gH @ yb; 0T where G is a third-order tensor, and s = (0, smom T ,0)T . 2.2. Treatment of discontinuous bathymetry

In its present form, the coupled system (3) presents severalchallenges when attempting to resolve all components of w simul-taneously. This is partly rooted in the fact that transport of sedi-ment is a non-passive quantity; two-way coupling results sincechanges in bathymetry directly affect both the elevation and owrate and vice versa. The more serious issue, however, is the pres-ence of the non-conservative products gH @ xb and gH @ yb in themomentum Eq. (1b) . Since the quantity gH cannot be expressedas the Jacobian of some tensor Q , the hyperbolic system (3) cannotbe written in so-called divergence form:

@ t w r Q 0: 4The problem becomes particularly acute when b is discontinuous,which frequently occurs in many ocean, riverine, and estuarine sys-tems. This is because weak solutions to (3) do not exist in this case;the distributional derivatives are not dened at the discontinuities[14] . As a result, traditional discontinuous Galerkin (DG) methodscannot be applied to this system in a straightforward manner.

There are three options available to treat this shortcoming. Therst is to usethe fully non-conservative form of (3) . Indeed, by con-sidering the fully non-conservative form of the coupled system inone space dimension, adjusted characteristic speeds may beextracted and incorporated into the denition of, for example,the Local LaxFriedrichs (LLF) numerical ux, and the systemmay then be expressed in divergence form. However, in two spacedimensions, this approach fails. Moreover, this option is not veryattractive from a numerical point of view, since the numericalmethod may then possibly converge to non-physical (i.e. non-entropy-preserving) solutions [16] .

The second choice involves uncoupling the sediment transportEq. (2) from the SWE (1a) and (1b) , at least to some degree. Basedon physical arguments involving typical time scales for sedimentand seawater transport rates, this approach was previously takenin our model implementation [16] . The principal drawback to thisapproach is its lack of accuracy during instances when the timescales for transport of sediment andseawater become similar, suchas during the extreme events ultimately desired to simulate.

The remaining option is to formulate a special treatment for thenon-conservative products. This was done in previous work forone-dimensional problems in [28] and for two-phase ow prob-lems in [29] . An alternative treatment was proposed by Dal Masoet al. [26] and applied by Rhebergen et al. to one- and two-dimen-sional problems using the DG method with piecewise linear nite

element basis functions in [14] . It is this treatment that will be em-ployed here.

The ideaproposedby Dal Maso et al. is to regularize the solution w near discontinuities by connecting the interior and exteriortraces w (in) and w (ex) (assuming these are well-dened) with asmooth path function / : 0; 1 ! R 4 parameterized by s 2[0,1].This path / must satisfy the following properties:1. [End conditions] / (0) = w (in) and / (1) = w (ex) ,2. [Consistency] If

w(in) =

w(ex) , then

/(s

) = w

(in) ,

3. [Lipschitz] For each component / k of / , $ M > 0 such that

@ / k@ s

6 M wink wex

k

for a.e. s 2[0,1], and4. [Symmetry] In two or more space dimensions,/ s; win; wex / 1 s ; wex; win :Condition (4) states that the path must be symmetric with

respect to an interchange of the traces. This condition need notbe satised for one-dimensional problems [14,26] .

Dening the third-order tensor T asT @ wF G; 5Eq. (3) may be rewritten as

@ t w T r w s 0: 6With the coupled system expressed by (6) in this non-conservativeform, we may state the following theorem, due to Dal Maso et al.[26] , on the integration of non-conservative products, which aretreated as a bounded Borel measure:

Theorem 1. Assume that

1. The function w is bounded on its domainX

,2. The function w is of bounded variation on X ,3. Each entry of T is a locally bounded, measurable function that

maps Borel sets to Borel sets, and4. The domain X = X C [ X J [ X I with each subset disjoint, where X C

is the set where w is almost everywhere continuous, X J is the set of jumps in w , and X I is the set of irregular points (an example of which is given in Section 3.1 ).

Then there exists a unique, bounded vector of Borel measuresl : X ! R 4 such that 1. If B is a Borel subset of X C , then

l

B

Z B T r

wdk;

where k is the Borel measure,2. If B is a Borel subset of X J , then

l B Z B\ X J Z 1

0 T / @ /@ s ds n dH 2 ;

where n is the unit vector normal to B directed toward the regionwhere s = 1 (where w = w (ex) ), H 2 is the two-dimensional Hausdorff measure, and 3. If B is a Borel subset of X I , then

l B 0:Consult [26] for the proof. The second assumption is necessary

since it guarantees that w(in)

and w(ex)

are well-dened, and alsoensures that w admits only jump-type discontinuities on X .

C. Mirabito et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 189207 191
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3. Finite element discretization

3.1. DG formulation

In this section, we derive the DGformulationof (6) . We begin bydening appropriate nite element function spaces to be used inthe model. Next, we construct an unstructurednite element mesh

or grid fromX

. Then weapply Theorem 1 to integrate the non-con-servative product in (6) over the mesh. Finally, we describe thenumerical ux to be used on element boundaries.

Let X h = [ eX e be a nite element partition of X into a set of non-overlapping triangular elements X e. The solution w is approxi-mated in X h by w h residing in the nite element space

W h L10; T ; V h4 ;

where

V h fv 2 BV Xh : v jXe 2 P pXe 8Xe 2 Xhg;where BV (X h) denotes the space of functions of bounded variationon X h (required for Theorem 1 ), and P pXe denotes the space of polynomials of order p on the element X e. In this paper, we assume p is constant over the domain; see [16,20] for commentary onimplementing p-adaptivity.

The semi-discrete weak formulation of the coupled system isderived following the approach discussed in [13,14] . MultiplyingEq. (6) by a test function vh 2W h and integrating over X h gives

@ t wh ; v hL2Xh Z Xh v h dl sh ; v hL2Xh 0; 7where Theorem 1 implies that

dl T hr whdXC h on XC h ;

R 10 T h/ @ /@ s ds n dX J h on X J h ;0 on XI h :8>>>:

8

In this context, XC h represents the interiors of all elements X e; X

J h ,

the set of all element boundaries o X e; XI h , the set of all nodalpointsa set of Borel measure zero.

With dl dened by Eq. (8) , Eq. (7) becomes

0 Xe @ t wh T hr wh sh ; v h L2XeXeI Z

1

0 T h/ @ /@ s ds n ; v h L2ceI ;

where the terms are now written element-wise, and ceI denotes aninterior edge. Splitting T h using Eq. (5) results in0

Xe

@ t wh r Fh Ghr wh sh ; v h L2 Xe

XeI Z 1

0 Gh/ @ /@ s ds n ; v h L2 ceI : 9

Since the test function vh is double-valued on element bound-aries, an appropriate numerical ux ^ v h must be chosen. Thisnumerical ux is chosen in [14] such that the DG formulation withnon-conservative products reduces to the conservative DG formu-lation whenever Gh is the Jacobian of some tensor Q h. To this end,Rhebergen et al. [14] prove that

^ v h ff v hgg v inh v

exh2

satises this requirement. Thus, vh may be replaced with { { vh}} in

the boundary integrals of Eq. (9) , and the divergence term in theelement interior integral may be integrated by parts to obtain

0 Xe @ t wh Ghr wh sh ; v hL2Xe Xe Fh ;r v hL2XeXeI f nc ; s v htD EL2ceI XeI Z

1

0 Gh/ @ /@ s ds n ;ff v hgg L2ceI

XeB f nc ; v inhD EL2 ceB; 10where ceB represents a boundary edge, s t = ( )

(in)( )

(ex)representsthe so-called jump operator, and ^f nc is a numerical ux to bedened

later.The derivation thus far has been independent of the exact form

of the path / . However, with w h regularized near the elementboundaries in this manner, the value of the regularized non-con-servative product, and hence the value of the path integral, is ulti-mately dependent in the limit on the choice of path (see, forexample, [14] or [26] ). As a consequence, Eq. (10) cannot be furthersimplied until a suitable form of / is chosen. Rhebergen et al. in[14] investigated the effect on numerical solutions for polynomialpaths of various orders, as well as those chosen by Toumi in [30] .They found that the effect on the numerical solution is small,and that having a good numerical integration scheme to integrate

the path integral appears to be of greater importance [14] . Withthis result in mind, a linear path will be used in the implementa-tion and in all results presented in this paper:

/ s win

h s wex

h win

h :This choice of path allows for easy evaluation of the path integral:Dene

v nc Z 10 Gh/ @ /@ s ds n 0

g ffH hggsbh t n x g ffH hggsbh t n y

0

0BBB@1CCCA

:

Eq. (10) can then be rewritten as

0 Xe @ t wh Ghr wh sh ; v hL2Xe Xe Fh ; r

v hL2Xe

XeI f nc ; s v h tD EL2ceI v nc ;ff v hggh iL2ceI XeB f nc ; v inhD EL2ceB:

To nalize the derivation, it remains to dene an appropriate(stable) numerical ux and merge the boundary integrals. As dis-cussed in [14] , the form of this ux may be thought of as a sumof an average term of Fh, a stabilizing viscous term involving s w ht ,and an additional stabilizing term involving vnc , the value of thepath integral. We dene

f nc f T nc ;hydro ; f nc ;sed T

;

where the numerical ux for the hydrodynamic part of the system(Eqs. (1a) and (1b) ) is dened, following [14] , as

f nc ;hydro Finh n 12 v nc S

in > 0;f HLL S

inS ex2 S ex S in

v nc S in < 0 < S ex;

Fexh n 12 v nc S ex < 0;

8>>>>>:11

where

S in min uin

h n ffiffiffiffiffiffiffi gH inhq ; uexh n ffiffiffiffi gH exhq ;S

ex

max uin

h n ffiffiffiffiffiffiffi gH in

hq ;uex

h n ffiffiffiffi gH ex

hq



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are the truncated characteristic speeds (found by considering theeigenvalues of the hydrodynamic part of T ), and

f HLL S exFinh S

inFexh nS ex S

in S inS exs wh t

S ex S in

is the HartenLaxvan Leer (HLL) ux dened in [31] . It should benoted that for most problems in shallow water hydrodynamics,the Froude number rarely exceeds 0.2, and therefore only the sec-ond case of Eq. (11) is frequently encountered [32] .

For thesediment transport Eq. (2) , an upwind ux is used, basedupon the assumption that transport is always in the direction of the ow. This choice of numerical ux is known to work well inpractice [16] . It is dened as

f nc ;sed qinb;h n u Roeh n P 0;qexb;h n u Roeh n < 0;

8 1 ; n2 > 1; n1 n2 < 0g:

Substituting the expansions (13) and (14) into Eq. (12) yields analgebraic linear system of ordinary differential equations

Xe Me d W h;edt Xe R h;e; 15where Me is the N N elemental mass matrix dened as

M e lm Z Xe /l x; y/ m x; ydXe;

the N 4 matrix W h ,e contains the degrees of freedom, and theN 4 matrix R h ,e denotes the remaining terms on the right-handside of Eq. (12) .

In order to solve the system (15) for W h ,e, we invert Me, andthen applya time steppingscheme. Usingan L2 b X e-orthogonalba-sis reduces the mass matrix M e to a diagonal one, which may betrivially inverted. For this reason, a hierarchical basis with theL2 b X e-orthogonality property proposed by Dubiner in [33] hasbeen implemented.

The time stepping scheme utilized is an explicit strong-stabilitypreserving (SSP) s-stage, kth-order RungeKutta scheme, wherethe pair ( s,k) is one of (1,1), ( s,2) with s P 2, ( s,3) with 3 6 s 6 8,

or ( s,4) with 5 6 s 6 8. The SSP RK scheme is designed so that if the forward Euler method is stable under a given semi-norm andCourantFriedrichsLewy (CFL) condition, then the higher-orderscheme remains stable under the same semi-norm, but perhaps adifferent CFL condition [16] . These methods also possess the desir-able total-variation-diminishing (TVD) property. This time step-ping method takes the following general form:

Algorithm 1 (SSP RK scheme). For each time step from t n to t n+1,

1. Set W 0h;e W h;et n.2. For each stage i = 1, 2, . . . , s, set

W ih;e

Xi

j

1

a ij W j 1h;e bijD t M1

e R h;e W j 1h;e ; t n d jD t

n o:

3. Set W h;et n1 W sh;e .

In the algorithm above, d j P j 1k1c j 1;k is a vector of length s of time-lagging parameters, and the s s matrices a , b , and c areopti-

mized parameters with respect to maximization of the CFL num-ber. They satisfy the TVD constraints

a ij P 0;

Xi

j1a ij 1; and

cij b ij

Xi 1

k

j

ckja i;k1 :

Tobe clear, it shouldbenotedthat Algorithm1 must beperformedforeach degree of freedom on every element X e. Refer to [16,18,34] forfurther details on SSP RK methods, and to [20] for values of a and b .

We conclude this section with someremarks on the use of slopelimiting. To eliminate local overshoots and undershoots, four dif-ferent choices of slope limiters are implemented as post-process-ing routines for W h ,e(t n+1). However, we only employ two of these here, since the others are notorious for degrading the solu-tion to rst-order accuracy at local extrema (effectively p =0 inthese regions), and we wish to test our model with p > 1 in theexamples listed in Section 4. Thus, when p = 1, the limiting proce-dure formulated by Cockburn and Shu in [35] is used if needed.This is the case for the entire morphodynamic system in Section

4.3, and for the sediment transport equation in Section 4.4. Butfor p > 1, it becomes necessary to make use of a higher-order slope



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limiting procedure. For this paper, we implement the algorithmoutlined in the work by Xu et al. in [36] , which is a local limitingprocedure based on the application of a WENO-type limiter tothe coefcients of the higher-order Taylor expansion about the ele-ment barycenter. It has been shown to work well in practice [36] .For the sake of brevity, we shall not expound upon the implemen-tation details here, but instead we refer the interested reader toSections 2.1, 2.2, 2.3, and 2.5 of [36] . In this paper, we found thatit was only necessary to apply the procedure of Xu et al. to the sed-iment transport equation for the test cases in Sections 4.2 and 4.4 ,and to the entire system for the case described in Section 4.3 . Forthe remaining test cases, we found that no limiting was necessaryin order to preserve numerical stability.

4. Numerical examples and results

In this section, numerical examples and results using our modelare presented. This method is applied to four problems: an evolv-ing dune, a converging channel, scouring around a bridge pier, andan idealized inlet. These simulations were run in parallel (with theexception of the rst test case) on the Institute for ComputationalEngineering and Sciences (ICES) bevo2 cluster, as well as on theTexas Advanced Computing Centers lonestar cluster [37] .

4.1. Convergence study: evolution of a dune

In this test case, a problem posed by Exner [4] in 1925 is exam-ined. This problem considers the evolution of an initially symmet-ric mound, or dune, which is subjected to steady, unidirectionalow under the assumption of a rigid-lid (which means that f is as-sumed constant in space and time). Solving this problem allows usto test the sediment transport model independently of the hydro-dynamics by comparing the numerical solution to a classical onederived in [4] for the purpose of verifying h- and p -convergencerates. It also allows us to verify and expand upon the results pre-sented in [16] for second- and third-order RKDG schemes.

To properly compare the numerical solution to the exact solu-tion, a simpler model must be used. Exners model uncouples thesediment transport equation from the SWE by specifying a unidi-rectional ow velocity as

u q f H

q f f b

;

v 0;where q f is the (non-negative) constant ow discharge rate. Notethat under the rigid-lid assumption, f is now assumed constantthroughout the domain X = ( k/2, k/2) 2. Also, it is assumed thatq b Akuk

n2 ; 0

T , where A is a non-negative constant and n = 1.When e0 = 0, Eq. (2) reduces to

@ b@ t @ @ x Aq f f b 0: 16Let b(0, x, y) = b0( x) be the initial data, which is assumed to besufciently smooth. Then the exact solution derived in [4] is givenimplicitly by

bt ; x; y b0 x ct ; 17a c b

Aq f

f b2 ; 17b

where c denotes the speed at which the top of the dune propagates.This is the same as the characteristic speed.

For this problem, the initial condition is given by

b0 x A0 A1 cos 2p x

k ; 18

with A0 = 2, A1 = 1, and k = 20. Also, we take f = 0 and Aq f = 1 in Eq.(17b) . As regards the boundary conditions, the solution is assumedperiodic in the x-direction over X , so that b(t , k/2, y) = b(t ,k/2, y),with no normal ow at y = k/2.

Nonlinear hyperbolic conservation laws have the well-knownproperty that solutions typically develop very steep gradientsandeventually discontinuities or shocks. This behavior can be seenin this problem when performing a rudimentary characteristicanalysis. Eq. (16) may be written as

@ b@ t

@ f @ x 0;

with f = Aq f /(f + b), and carrying out the partial derivatives givesthe non-conservative form

@ b@ t c b x

@ b@ x 0;

where c (b( x)) = @ f /@ b. Then it can be seen that the breaking timewhen the shock (or sediment bore) forms is given by

t b minnd

dnc b0n

1

5 5 ffiffiffi7p 3

4p ffiffiffiffiffiffi2 ffiffiffi7p 4p 4 :5685 s :

In our study, four meshes denoted h1 through h4 are considered. Thecoarsest mesh h1 consists of 800 elements, arranged according toFig. 2(a). The value of h , dened here as the length of an elementin the x-direction, is 1 m. Mesh h2 is a 1:4 renement of h 1, withh = 1/2 m and 3200elements (see Fig. 2(b)). This process is repeatedto generate meshes h3 and h4.

The numerical model is then run until the breaking time t b isreached. The time step size is set small enough so that the spatialerrors dominate the overall error. The DG solution bh is computedusing piecewise constant, linear, quadratic, and cubic basis func-tions ( p = 0,1,2, and 3) with the SSP (1,1), SSP (2,2), SSP (3,3),and SSP (5,4) time stepping schemes, respectively, and the errorsare computed by comparing the approximate solution bh to the ex-act solution given by Eqs. (17a) and (17b) . The h-convergence re-sults using the L1 (

X h)-norm at t =2 s and t = 4 s are shown inFig. 3(a) and Fig. 4(a), respectively, while the p-convergence resultsusing mesh h4 and the L1 (

X h)-norm at t = 2 s and t = 4 s are shownin Figs. 3(b) and Fig. 4(b), respectively.

At t = 2 s, the solution is still smooth, as this time is well beforethe time of formation of the shock. Thus, the theoretical h-conver-gence rate of p + 1 is very nearly attained, as can clearly be seen inFig. 3(a) for the L1 (

X h)-norm. It should be noted that these ratesare also attained for the L1(X h)- and L2(X h)-norms, though theseresults are not shown here. For p-convergence, the theoreticalexponential rate is attained, and this is shown in Fig. 3(b).

However, for t = 4 s,which is muchcloser to the breakingtime t b,the h-convergence rates areeither seriouslydegraded (for thecases

of p = 0, 2, and 3) or obliterated (for the case p = 1) when the errorover the entire domain X h is under consideration. This is shown inFig. 4(a) for the L1 (

X h)-norm, and similar results were obtainedfor the L1(X h)- and L2(X h)-norms. In this case, the largest portionof the error has accumulated in the region {( x, y): x 2(4,6), y 2( 10,10)}, which corresponds to the region containing the steepestgradients in thesolution,and is also theregion containing theshockwhen t P t b. This issue demonstrates the need for further h-rene-ment inthis region, asthe results shownin Fig. 4(a)appearto displaypre-asymptotic behavior. However, when accuracy is consideredonly in the smooth region of the solution, taken here as

Xsm f x; y : x 2 10 ; 4; y 2 10 ; 10g;it can be seen in Table 1 that nearly full order of convergence is

restored for thecases of p = 0 and p = 1, while theconvergence ratesfor the cases of p = 2 and p = 3, though not quite optimal, are much

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improved, and appear to be tending towards the theoretical ratespending further h-renement. This veries that the rst- throughfourth-order RKDG methods maintain their accuracy and theoreti-cal order of convergence away from theregioncontaining steepgra-dients and shocks.

In Fig. 4(b), we see that the p-convergence rate at t = 4 s doesnot appear to seriously degrade until p = 3. Since the results onmesh h4 were used here, the effects of the value of h being in thepre-asymptotic regime are mitigated; indeed, the convergence rate

is further degraded when using mesh h3, and no convergence isobserved when using meshes h2 and h1 (not shown here). Despite

these shortcomings, however, these results show that the modelmaintains its accuracy in p when h is small enough.

4.2. A converging channel

In [4] , Exner extended the model presented for the previous testcase to account for channels of varying width with vertical walls.He considered a converging or bottleneck channel with an initially

at bed. This is the computational domain considered in this testcase, and it is shown in Fig. 5. The channel considered is 2 km long,500 m wide at the eastern and western ends, and narrows to awidth of 250 m at the center.

Rather than considering the evolution of the bed in a one-dimensional setting, as Exner did, we utilize the full morphologicalsystem to consider it in a two-dimensional setting. This allowssimultaneously examining both the evolution of the bed andchanges in thevelocity andowrate patterns over a 90-day period.For this case, a no normal ow boundary condition is assumed forthe northern and southern channel walls, while a xed inow rateof 5 m 2/s is imposed at the western end of the channel. This ratewas chosen so that the ow speed is 0.5 m/s at either end of thechannel, with a maximum speed of approximately 1 m/s in the

throat of the channel. At theeastern boundary, theelevation is heldxed at f = 0, chosen to enforce mass conservation. The bed is ini-

Fig. 2. Computational meshes (a) h1 and (b) h2 showing the h-renement patternutilized in the dune evolution test case. Mesh h3 (not shown) is a 1:4 renement of mesh h2, and mesh h4 (also not shown) is a 1:4 renement of mesh h3.

Fig. 3. L1 (X h) convergence rates at t =2 s.



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tially at, with a bathymetric depth of 10 m. The bottom is also

non-porous ( e0 = 0), and the sediment density is taken asq s = 2000 kg/m 3, while the median grain size is d50 = 0.2 mm. For

the purpose of calculating qb, the seawater density is taken asq = 1024.763kg/m 3, which is the value given by the UNESCO1980 International Equation of State (IES-80; see [38] for details)with temperature T = 20 C andsalinity S = 35 psu. For the hydrody-namics, nonlinear bottom friction given by sbf = c f kuk2/H is consid-ered, where c f is the (dimensionless) friction coefcient or bedroughness, taken here to be 0.0025.

For purposes of comparison, this test case is run using two dif-ferent grids. The rst is a coarse mesh, shown in Fig. 5(a), and thesecond is a ner-scale mesh, shown in Fig. 5(b). For each of thesegrids, numerical realizations of w h using piecewise constant, linear,

and quadratic basis functions are considered. For the case of piece-wise constant basis functions, time step sizes of 1.5 s (coarse grid)and 0.25 s (ne grid) are used with the Forward Euler scheme SSP(1,1); for piecewise linear basis functions, 0.75 s and 0.1 s, respec-tively, with SSP (2,2); for piecewise quadratics, 0.5 s and 0.1 s,respectively, with SSP (3, 3). Recall from Section 3.2 that a high-er-order slope limiter described in [36] is applied when p = 2,which allows us to take slightly larger time step sizes than usual,though still well below the theoretical threshold given by the CFL condition. For example, using the estimate for the CFL restrictiongiven by

D t 6 minXe

heu n

ffiffiffiffiffiffi gH

p e

12 p 1

0B@

1CA

; 19where he is dened here to be the longest (elemental) edge length,and 1/(2 p + 1) is an estimate of the CFL number [16] , we obtainD t CFL 0.85 s for the coarse grid and D t CFL 0.15 s for the ne grid.In practice, however, time step sizes of approximately D t CFL /2 mustbe taken before the model becomes stable when not utilizing aslope limiter; the ability to take time step sizes larger than D t CFL /2 is thus a welcomed feature of this slope limiter.

Plots of the bathymetry for p = 0, 1, and 2 on the coarse mesh att = 90 days are shown in Fig. 6, while Fig. 7 shows the correspond-ing velocity magnitude. In Fig. 6, it can be seen that the bottomundergoes erosion in the converging part of the channel becausethe ow is accelerating there (shown in the corresponding guresfor kuk2). Conversely, sediment accretion canbe seen in thediverg-ing part of the channel since the ow speed is decreasing there. Theerosion and accretion patterns are not uniform across the width of

Fig. 4. L1 (X h) convergence rates at t =4 s.

Table 1

Comparison of h -convergence rates at t = 4 s on the entire discretized domain X h andon the smooth region X sm only.

p Mesh L1 convergence rates

in X h in X sm

0 h1 h2 0.6305 0.9762h3 0.6641 0.9892h4 0.7001 0.9967

1 h1 h2 0.1522 1.8339h3 1.4000 1.7602h4 0.7128 1.8740

2 h1 h2 1.8513 1.8512h3 1.0743 2.3752h4 2.2383 2.6923

3 h1 h2 1.5706 3.4398h3 2.2724 3.1037h4 2.0180 3.5808

Fig. 5. Unstructured computational meshes used in the converging channel testcase. The ratio h coarse /hne 4.88, where h coarse and h ne are dened here as thelongest edge lengths in the coarse and ne grids, respectively.



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the channel, however, as it can be observed in Fig. 7 that the veloc-ity is nonuniform across the channel width; it is slightly highernear the center. Since the sediment transport rate q b is a functionof kuk

32 , the magnitude of the erosion and accretion is seen to vary

considerably across the channel width.The numerical solution exhibits qualitatively similar behavior

on the ne-scale mesh, except with generally better resolution.Consider the bathymetric proles shown in Figs. 6 and 8 . For thepiecewise constant case, notice that the ne-scale solution reectshigher sediment mass transport rates. For the cases of p = 1 and

p = 2, differences in resolution are more pronounced: the solutionon the ne mesh appears to better account for the land boundaries,and the resolution in the transition regions (on the upstream edgeof the scouring region, between the erosion and deposition zones,and just downstream of the deposition region) is much sharper.The piecewise quadratic solution on the ne mesh ( Fig. 8(c)) moresharply captures the transport behavior in the center of the chan-nel; the cross-stream prole is not as well-resolved on the coarsemesh, especially in the accretion zone.

Now consider the velocity proles shown in Figs. 7 and 9 . Forthe case of p = 0, the ne-scale solution shows a slightly largerspeed gradient, especially near the boundaries, explaining the dif-ferences in bh between Fig. 6(a) and Fig. 8(a) as a result of the dif-fering transport rates. For the cases of p = 1 and p =2, one

immediately spots the differences in the ow behavior near theland boundaries in the channel throat when comparing them with

the piecewise constant results. The increase in velocity from thechannel center to the boundary region is more well-dened; thisfeature appears to be smeared in Fig. 7(b) and (c). The highestspeeds occur nearer to the boundaries in the throat, and the veloc-ity gradient downstream of the throat is much higher compared tothe coarse-grid solution.

When p P 1, a long, thin boundary layer is seen downstream of the channel throat. Its appearance is consistent with the resultsshown in [13] , and may bepartly due to the effect of the no normalow BC there (but free tangential slip is allowed). However, we

note that by comparing Fig. 7(b) and Fig. 9(b), its thickness appearsto depend on the grid size: as h ? 0, this layer becomes thinner,and appears to be conned only to those elements next to the landboundary.

Comparing the DG solutions for b and kuk2 across the differentvalues of p, we immediately notice that the p = 0 approximation of b only supercially captures the scouring and accretion behavior;the computed sediment transport rate in the converging regionappears too high, while it is too low along the land boundaries.Thus, some numerical diffusion is evident here; the p = 0approximation of b appears to be overly diffusive. Increasing p dra-matically improves the resolution of bh, especially near the bound-aries, in particular for p = 2, where the solution appears to sharplyresolve the subgrid-scale features in the transition region. Fig. 6(b)

and Fig. 8(c) show that the higher-order slope limiter suppressesoscillations near the land boundaries in the throat of the channel

Fig. 6. Bathymetry (m) after 90 days on the coarse mesh. Fig. 7. Flow speed (m/s) after 90 days on the coarse mesh.



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when p = 2. Turning our attention to kuk2, we nd a similar prob-lem of resolution when p = 0: the cross-stream velocity prole ap-pears to be nearly uniform in the throat of the channel, and themodel fails to capture the behavior of the solution near the landboundaries in the channel throat; the predicted speed still appearsto be too low.

Based on the issues pointed out thus far, it should be noted thatthecoarse-gridsolutionsdo notseemto besatisfactory foranyvalueof p 2{0,1,2} when comparedwith some results shown in theliter-ature [13,25] . Results obtained using the ne-scale grid arestill notwell-resolvedwhen p = 0,but aregenerallyverywell-resolvedwhen p P 1. These observations, especially the nal one, underscore theneed for taking p of sufcient order in the DG approximations.

It is important to observe the interaction between the bathy-metric and velocity proles as both evolve in time: as the bed iseroding in the converging part of the channel, the total depth H in-creases, which in turn acts to decrease kuk2. Conversely, as amound begins to form on the bed in the diverging part of the chan-nel, H decreases, causing an increase in kuk2 in this region. Thisongoing (coupled) process results in the entire velocity proleslowly shifting slightly downstream with time. These results areshown in Figs. 10 and 11 .

Lastly, it should be noted that the results shown in Figs. 69 all

compare well qualitatively with an analytical solution derived byExner in [4] for a problem with similar geometry, with results

shown in Tassi et al. [13] for this problem posed in a non-dimen-sional setting, and with thecomputed results presentedin Kubatkoet al. [25] using a combined CG-DG model (for p = 1).

4.3. Scouring around a Bridge Pier

In this section, another well-known and well-studied problemin coastal engineering is examined: scouring, or bed erosion,around a vertical, cylindrical bridge pier, such as those commonlyseen on roadway and railway bridges across rivers and estuaries.

Applications of these problems to coastal engineering activitiesmotivate our model testing here, as scouring has been cited asone of the most common causes of bridge failures in these regions[8,39] . We also wish to verify that our model accurately depicts themajor ow features associated with the scouring process, at leastin a qualitative sense. This problem would be more accurately sim-ulated by a three-dimensional NavierStokes model, but an inter-esting question is to determine which features can be capturedby a hydrostatic, depth-averaged model.

In this problem, an initially at bed is subjected to a uniform,unidirectional ow originating from the western (upstream)boundary. This free stream or upstream velocity is denoted asu 1 = (u1 ,0)

T . The domain under study in this test case is a squareof side length 30 m with a circular island of diameter D = 2 m posi-tionedat the origin. However, since the true solution w is symmet-ric about the x-axis, our domain is chosen as

Fig. 8. Bathymetry (m) after t = 90 days on the ne mesh.Fig. 9. Flow speed (m/s) after t = 90 days on the ne mesh.



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X f x; y 2R 2 : x 2 15 ; 15; y 2 0; 15; x2 y2 > 1gin order to save computational time. Two computational meshesshall be used for this test case for comparison. These are shown inFig. 12 . The bed is initially at with a depth of 1 m. Boundary con-ditions are similar to those for the previous test case, and are as fol-lows: at x = 15, the velocity is xed at u1 = 0.2 m/s. At the bridgepile boundary, a no normal ow boundary condition is imposed. Nonormal ow is also imposed at the top ( y = 15) and along the axis of symmetry ( y = 0). At x = 15, the free surface elevation is specied asf = 0. Values for e0, q s, d50 , q , and c f are the same as for the test case

in Section 4.2 . However, in order to save computational time, thesediment transport rates are articially inated by a factor of 1000 so that the bed evolution may be examined after 1 day, wellafter the ow becomes fully-developed.

Following the pattern begun with the previous test cases, weseek whjXe 2 P pXe for p = 0, 1, and 2 for both meshes. When p =0, the time step sizes are set to 0.025 s (coarse mesh) and0.01 s (ne mesh) and SSP (1,1) is employed; p = 1, 0.01s and0.005 s, respectively, with SSP (2,2); p = 2, 0.0075 s and 0.0025 s,respectively, with SSP (3,3). Recall from Section 3.2 that the slopelimiter described in [35] is applied to all components of w h in orderto control and/or suppress oscillations when p = 1. For the piece-wise quadratic case, the WENO-type limiter described in [36] isused, again for the purpose of enforcing some form of TVB stability.

For this problem, experimental results show that the basicscouring process and ow prole evolution may be summarized

as follows. The initial approach ow splits in the horizontal direc-tion when reaching the pier into an upper and lower ow, leaving astagnation point at the head of the pier (the point ( 1,0) in ourdomain). A corresponding stagnation pressure results; uidapproaching the stagnation point decelerates, causing some pileup(and thus an increase in f ) [39] and a bow wave, and then acceler-ates along the sides of the pier [40] . Thefully-developed owspeedalong the pier perimeter reaches its maximum at an angle h 75from the pier head, but then separates in the region 90 6 h 6 120as a result of the acceleration along the sides [41] , and a long wakeresults. Meanwhile, scouring of the bed is initiated near h 75 ,

closely corresponding to the point where the (attached) accelerat-ing ow reaches its maximum speed [41] . Sediment accretion oc-curs just downstream of the pier, and the mound slowly driftsdownstream with time because of the growing scour hole [41] .

However, it should be noted that in addition to this, the ap-proach ow also splits in the vertical direction at the upstreamstagnation point; the up-ow contributes to the formation of thebow wave, while the down-ow forms a vertical eddy or vortex[40] . Becauseof accelerating ow around the pile, a horseshoe vor-tex system is subsequently formed, andthis has been deemed to bethemain scouring agent upstreamof thepier [40,41] . Thus, in time,the scour hole advances upstream, eventually becoming U-shaped[40] . We note that this latter process is inherently a three-dimen-sional effect, and consequently we cannot hope to properly simu-

late it with our two-dimensional model. We proceed with thiscaveat in mind.

Fig. 10. Bathymetry (m) at 30-day intervals on the ne mesh with p = 2.Fig. 11. Flow speed (m/s) at 30-day intervals on the ne mesh with p = 2.



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The resulting bed proles are shown in Fig. 13 for the coarse-scale mesh and in Fig. 15 for the ne-scale mesh. The correspond-ing velocity proles are displayed in Fig. 14 for the coarse-scalemesh and in Fig. 16 for the ne-scale mesh. In all results, the loca-tions of the stagnation points, scouring initiation points, points of maximum ow speed along the perimeter, and ow separationpoints appear to be in good agreement with the results presentedin [39,41] .

In Figs. 13 and 15 , it can be seen that scouring emanates from apoint on the pier perimeter near h = 75 , which agrees with exper-imental data in [41] , and then is deposited just downstream of thepile, as should be expected. However, the extent of the scouringvaries considerably between the coarse- and ne-scale meshes,especially when p P 1; h-renement near the bridge pier appearsto play a signicant role here in the accurate calculation of the sed-iment transport rate. Also, when p P 1, the resolution of the scour

hole seems to be slightly improved, and its extent appears to be re-duced. The same may be observed with the corresponding velocityproles: the wake appears slightly sharper and more compact(although the ow seems to separate slightly too far downstreamcompared to the coarse-scale solution, which shows separationaround h = 120 ), as does the ow near the upstream stagnationpoint.

By closely observing the differences in the scouring and accre-tion proles when p P 1, the following may be noted. When p = 1, differences between the coarse- and ne-scale solutions arequite apparent; the coarse-grid scour hole appears smeared, andsimply emanates outward at h 75 , while the ne-grid holeexhibits a downstream tail. Its visual appearance is similar to thatof the p = 2 coarse-grid solution. While the corresponding ne-grid

solution for p = 2 shown in Fig. 16(c) displays some additionalstructure, it is not substantially qualitatively different. Conse-

quently, the piecewise quadratic approximations to b may possiblybe showing some signs of convergence (although additional trialswould be needed to verify this).

Many differences in the bed evolution can be remarked uponamong the cases for p = 0, 1, and 2. For the piecewise constant case,

Fig. 12. Unstructured computational meshes used for the bridge pier scouring testcase. The ne-scale mesh (b) is a uniform 1:4 renement of the coarse-scale mesh(a).

Fig. 13. Bed proles after 1 day on the coarse mesh.



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notice that the eroding and accretion regions extend nearly two-thirds of the way to the domain boundary. This is clearly unrealis-tic, since both experimental and analytical investigations suggest

that the boundary of the scour hole lies much closer to the pile[40] . Thus, the RKDG scheme with p = 0 seems to be much too dif-

fusive, and this demonstrates the need for increasing p to sufcientorder. Also, the maximum scour and sediment accumulations arefar less than the maxima attained for the other two cases, possiblyindicating that the calculated sediment transport rates may be too

low. This agrees with our observations in Section 4.2 . When p isincreased to one, a more realistic result emerges, as the area of

Fig. 14. Velocity proles after 1 day on the coarse mesh.

Fig. 15. Bed proles after 1 day on the ne mesh.



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erosion and deposition is constrained to a distance not exceedingapproximately 2 D. The higher scouring and accretion rates maybe attributed to the larger velocity gradients seen in Figs. 14(b)and Fig. 16 (b), especially in the wake, whose width is now much

smaller; it is not smeared. The piecewise quadratic solutionsappear even less diffusive, even with the use of the higher-order

slope limiter: an even larger amount of sediment pick-up occursin a still smaller scour hole located closer to the pier comparedwith the p = 1 case, and the ends of the U-shape attained by thescour hole are more evident. The wake and upstream stagnationregions are more compact as well. Additionally, observe that theaccretion region emanates outward at a larger angle h as p is in-creased. It should be noted, however, that the general lack of diffu-sion seen in these results for

pP 1 is primarily a numerical

phenomenon, and it is a consequence of our decision to ignorethe diffusive terms in Eqs. (1a), (1b) and (2) ; some physical diffu-sion will be seen in reality.

Lastly, we note that examination of the elevation prole for allmodel runs revealed strong numerical evidence of a bow wave, asvalues of f were elevated near the upstream stagnation pointwhere seawater pileup is expected. However, for the sake of brev-ity, we do not show these results here.

4.4. An idealized inlet

We now turn our attention toward a coastal modeling applica-tion. In this test case, an idealization of a typical coastline, channel,and bay or estuary is considered. For this case, we wish to ensurethat the model properly captures the twin eddy formation in theback bayduringan incoming (ood) tide, thecorresponding forma-tion in the ocean during an outgoing (ebb) tide, the formation of anebb shoal in the ocean off the ends of the jetties, and the formationof a corresponding ood shoal in the back bay. Ebb and ood shoalformation is very common in both coastal inlets and river deltas,and they form as a result of sediment accretion caused by deceler-ating ow exiting the channel during the ebb and ood tides,respectively. Though the time scales are generally on the order of months to many years (assuming fair weather), this process, leftunchecked, can severely alter the ow patterns of an inlet and dis-rupt shipping operations, and the ensuing dredging operation maybe very expensive. Thus, analysis of these systems serves an impor-tant purpose. An example of an ebb and ood shoal is shown inFig. 17 .

The domain under consideration here is shown in Fig. 18 . It is asimplied, or idealized, inlet containing a channel 300 m wide and525 m long which connects a rectangular section (stretching 3 kmnorth to south and 2.475 km out to sea) of open ocean to the westwith a rectangular back bay (spanning 3 km north to south and1.5 km inland) to the east. At the western end of the channel aretwin jetties 50 m wide and 225m long. The initial bathymetry inthe back bay and channel is at and measures 5 m, while the bot-tom slopes linearly in the open ocean to the west, measuring 19 mat the western open ocean boundary. The water in the sea, channel,and back bay is initially tranquil.

The boundary conditions for this problem are as follows: a nonormal ow condition is imposed at the northern, southern, and

eastern edges of the open ocean, as well as at the jetties, channelwalls, and all edges of the back bay. On the western open oceanboundary, we specify a spatially uniform periodic ow rate, withan amplitude of q (ex) = (0.75,0) T m 2/s, and a frequency correspond-ing to that of the M 2 tide (period 12 h25 m 14 s). This amplitude waschosen so that the maximum ow speed in the channel is approx-imately 1 m/s.

This test case is solved over a period of 10 days using piecewiseconstant, linear, and quadratic basis functions for the purpose of comparison. For the piecewise constant approximation, a timestepsize of 1.25 s with SSP (1,1) is used; for the piecewise linearapproximation, 0.5 s with SSP (2,2); for the piecewise quadraticapproximation, 0.25 s with SSP (3,3). Recall once more from Sec-tion 3.2 that for the sediment transport equation, the slope limiter

of Cockburn and Shu [35] is applied when p = 1, and the higher-or-der slope limiting procedure of Xu et al. [36] is applied when p = 2.

Fig. 16. Velocity proles after 1 day on the ne mesh.



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Thebed remains non-porous, with the same values for d50 and q asin Section 4.2. However, the grain density is now increased to2650 kg/m 3. The nonlinear bottom friction coefcient c f is in-creased to 0.003 for this case. Additionally, the sediment transportrates were magnied 50 times in order to speed up the bed evolu-tion process. This was done in the interest of saving computationaltime.

Fig. 19 (a), (b), and (c) show plots of the piecewise constant, lin-ear, and quadratic approximations of b after 10days, respectively,in the vicinity of the channel. The other areas of the domain expe-rience negligible amounts of sediment transport, and so are not

shown here; the most interesting ow features are found in thechannel region. The most prominent of these is the formation of dual scour holes in the center of the channelone located at theentrance to the back bay, the other just off the ends of the twin jet-

Fig. 17. A photographic description of ebb and ood shoal formation in ShinnecockInlet, NY, USA. Photograph courtesy NOAA Coastal Services Center.

Fig. 18. Computational domain of the idealized inlet system containing 4374elements. The initial bathymetry(m) is shown as well. The jetties are225 m long by50 m wide.

Fig. 19. Bathymetry (m) in the channel region at t = 10days.



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ties. These scour holes are caused primarily by converging acceler-ating ow during the incoming tide (for the hole near the jetties)

and during the outgoing tide (for the hole at the bay entrance). Thisdual scour hole formation is consistent with what is observed in

real channel-inlet systems, such as Shinnecock Inlet, NY, wherebathymetric soundings reveal dual scour holes in approximately

the same locations as described above (though these are slightlyoff-center because the channel is not straight, as can be seen in

Fig. 20. Velocity proles during approximate maximum ebb. Fig. 21. Velocity proles during approximate maximum ood.



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Fig. 17 ). The jagged shape of the scour holes is a direct result of therelative coarseness of the mesh in this region (see Fig. 18 ); this gridsize was chosen as a compromise between resolution and compu-tational time.

Notice also in Fig. 19(b) and (c) that sediment accretion occursin the center of the channel. This phenomenon is due to the slightlynegativevelocity gradient existing there for most of the durationof the

M 2 tidal cycle. See Figs. 20 and 21 , and observe the channel

center. These results suggest a mechanism by which channelscould ll in over longer time periods and possibly block the ow,posing a hazard to shipping interests, however, this requires fur-ther investigation. Finally, we notice the formation of ebb andoodshoals beyond the scour holes, which are also caused in part by thenegative velocity gradients in these regions during the ebb andood tides, respectively.

Comparing the results across the different values of p, we re-mark that the piecewise constant approximation to b once againfails to capture the full extent of the induced bed evolution: thescour hole depths are approximately 9 cm and 7 cm at the jettiesand back end, respectively, compared with 49 cm and 36 cm forthe piecewise linear case and approximately 55 cm (both ends)for the piecewise quadratic case. It should be noted, however, thatvalues for the scour hole depths for p P 1 are highly dependent onthe use of the slope limiter; it should be made clear that withoutthe use of a slope limiter, runaway scouring will eventually occur,and the resulting scour hole depths will become non-physicallylarge, leading to numerical instability. In any case, the depths weresubstantially higher than those obtained when p = 0.

The maximum speed in the channel is approximately 0.2 m/slower when p = 0 compared with p > 0, and since the transport ratedepends on kuk

32 , the induced transport rate of sediment is lower.

Low deposition rates for this case result in scant evidence of ebband ood shoal formation, and of accretion in the channel center.These low observed rates of sediment transport compared to high-er-order approximations are again consistent with what was seenin Sections 4.2 and 4.3 .

Seeking a higher-order approximation of w yields a dramaticimprovement in resolution, as numerical evidence for ebb andood shoal formation is clearer, and sediment accretion in the cen-ter is more easily seen. The solution for the case of p = 2 appears tobe even less diffusive; the scour holes are now very well-dened,as is the accretion in the channel center.

For this test case, as alluded to earlier, it is worthwhile to exam-ine the velocity proles at two different timesduring the approx-imate time of maximum ebb, which is the time in which theseaward current associated with the outgoing tide is the strongest,and during the approximate time of maximum ood, when theincoming tidal current is the strongest. Fig. 20 shows the ow pro-le for p = 0, 1, and 2 at maximum ebb. Immediately it is seen thatthe numerical solution for the piecewise constant approximation

to kuk2 is unrealistically simple; the model cannot resolve the dualeddies that form in the ocean as a result of interaction with the jet-ties. Even within the channel, the grid- and subgrid-scale ow fea-tures are not well-resolved. This changes drastically when p P 1,as Fig. 20(b) reveals two sets of dual eddies. One of these is a pairof jetty-scale eddies appearing in the ocean near the leading edgeof the main seaward current coming from the channel. A pair of kilometer-scale eddies in the back bay is present as well, and theseare residual eddies which contain remnants of the seawater trans-ported into the bayduring the previous incoming tide. Theysupplythe seawater mass for the return ow. The p = 2 solution appears togive slightly better resolution of these large-scale eddies. Noticealso that the solutions for p P 1 reveal much greater detail withinthechannel, especially near the scour holes, where small regions of

increased velocity are seen. In addition, long, thin tails of slow-moving seawater are present near the channel wallsa feature

not seen in the p = 0 case, but consistent with results shown in[16] . Grid- and subgrid-scale velocity gradients are better resolvedhere, too, specically in the piecewise quadratic case. All of theseobservations underscore the need to take p P 1 in order to obtaina reasonable ow prole.

The velocity proles for the opposite tidal phase are shown inFig. 21 . Similar remarks from the previous paragraph about thesolutions for the different values of

p apply here as well: the

numerical solution for p = 0 is still far too simplistic, while thepiecewise linear approximation only partially resolves the (nowclearly visible) dual eddies in the back bay, which appear to benearly fully resolved only when p = 2 (shown in Fig. 21(c)). Theseobservations and results are consistent with those in [16] . Note,however, that during the ood tide, a pair of unnatural residualkilometer-scale eddies remain in the ocean, which is likely a resultof the interaction between the outgoing current (originating fromthe channel) and the ow-specied western boundary: they likelyaredeected back into the domain anddo not dissipatequickly en-ough since Eqs. (1b) and (2) neglect diffusion terms. This not onlyhighlights the issues relating to numerical diffusion, or lack there-of, but also highlights the ongoing issue of proper specication of open ocean boundaryconditions in the framework of theDG meth-od, which were taken to be

bex bin;

H ex H in; and

qex 0:75 ; 0T

for this test case. Both issues need to be further investigated.As was the case during maximum ebb, the cross-stream ow

features are better resolved when p is increased to at least one,and they are very well-resolved when p is increased to two, whichsuccessfully captures the more complex features. Specically, con-sider the ow between the jetties: the maximal speed occurs attwo locations in this region, compared to just one when p = 1.

Obviously, this phenomenon occurs during the outgoing tide aswell, but is not quite as easily discerned in Fig. 20(c) because weare not quite at maximum ebb. Again, these results agree withthose presented in [16] , and the presenceof multiple local maximain kuk2 suggest the presence of multiple local maxima in b as well.This is indeed the case, and may be spotted in Fig. 19(c). Thus, thecorresponding bed prole may be thought of as possessing a dualdual scour hole arrangement, rather than simply a dual setup asseen in Fig. 19(b).

4.5. Parallel performance

We conclude this section with some brief remarks on the per-formance aspect of the model in parallel. The numerical results

presented in the previous three sections have focused on theimproved accuracy in w h with increasing p. However, thisimprovement in accuracy comes with the price of additional com-putational costs as p is increased. For example, increasing p fromzero to one triples the number of degrees of freedom, and, in thecase of the converging channel with the coarse grid, requires twiceas many time steps. Also, additional computational work is neededto evaluate the interior and edge integrals, as more quadraturepoints are required. We attempt to quantify the amount of extrawork required in the test case that follows.

Consider once again the converging channel test case describedin Section 4.2. For this trial, only the coarse grid is considered. Ourobjective is to measure the relative CPU times when the model isrun for p = 0, 1, and 2. We run the model using the same parame-

ters as described in Section 4.2 , except that the model is run for1 day, instead of 90 days. For each run, 4 processors were utilized,

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