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ARTICLE IN PRESSJID: APM [m3Gsc;October 22, 2015;13:26]

Applied Mathematical Modelling 000 (2015) 1–17

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Well-balanced central schemes for systems of shallow waterequations with wet and dry statesR. Touma∗Mathematics, Lebanese American University, Beirut, Lebanon

a r t i c l e i n f o

Article history:Received 23 February 2015Revised 11 August 2015Accepted 23 September 2015Available online xxx

Keywords:Unstaggered central schemesWell-balanced schemesShallow water equationsSurface gradient methodWetting and dryingPositivity preserving schemes

a b s t r a c t

In this paper we propose a new well-balanced unstaggered central nite volume scheme forthe shallow water equations on variable bottom topographies, with wet and dry states. Basedon a special piecewise linear reconstruction of the cell-centered numerical solution and acareful discretization of the system of partial differential equations, the proposed numericalscheme ensures both a well-balanced discretization and the positivity requirement of the wa-ter height component. More precisely, the well-balanced requirement is fullled by followingthe surface gradient method, while the positivity requirement of the computed water heightcomponent is ensured by following a new technique specially designed for the unstaggeredcentral schemes.The developedscheme is thenvalidated andclassicalshallow waterequationproblems on variablebottom topographieswith wetand drystates aresuccessfully solved. Thereported numerical results conrm the potential and efficiency of the proposed method.

© 2015 Elsevier Inc. All rights reserved.

1. Introduction

Introducedby Nessyahuand Tadmor[1]in 1990, central schemes were meant tobe simpleand efficient toolsfor thenumericalsolution of systems of hyperbolic conservation laws. Based on the staggered Lax–Friedrichs scheme, the Nessyahu and Tadmor(NT) central scheme evolvesa piecewise linear numerical solution on two staggered grids andavoids the time consuming processof solving Riemann problems arising at the cell interfaces. Furthermore, second-order quadrature rules along with a gradientslimiting process result in a second-order accurate and oscillations-free nite volume method for general hyperbolic systems of conservation laws. Multidimensional extensions of the NT scheme on Cartesian grids [2–5] and unstructured grids [6–9] werelater developed and successfully applied to solve hyperbolic arising in aerodynamics and magnetohydrodynamics [5,10], as wellas balance law problems arising in hydrodynamics [11–13].

One main disadvantage of using central schemes remains in the fact that a dual staggered grid is required to evolve the nu-merical solution, and this leads to synchronization issueswhenever physical constraints are to be numerically forced, or if steadystates are to be handled in the case of balance laws. To overcome this problem, Jiang et al. [14] presented a rst adaptation of theNT scheme that evolves the numerical solution on a single grid and another adaptation (known as unstaggered central schemeUCS) followed in [13]. Based on a careful projection of the numerical solution obtained on the staggered cells, back onto theoriginal cells, the UCS method was successfully used to solve steady state problems arising in hydrodynamics, and ideal/shallowwater magnetohydrodynamics [15,16]. In this paper we develop a new well-balanced second-order accurate unstaggered centralschemefor thenumerical solution of shallow water equation(SWE) problemswith wet anddry states. TheSWE system describes

∗ Tel.: +9613834095.E-mail address: [email protected]

http://dx.doi.org/10.1016/j.apm.2015.09.073

S0307-904X(15)00607-1/© 2015 Elsevier Inc. All rights reserved.

Please cite this article as: R. Touma, Well-balanced central schemes for systems of shallow water equations with wet and drystates, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.09.073

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2 R. Touma/ Applied Mathematical Modelling 000 (2015) 1–17


the motion of a free surface incompressible uid under gravity, over a variable bottom topography. The SWE system is widelyaccepted as a mathematical model for the hydrodynamics of coastal oceans, simulation of ows in rivers, and can also be used tosimulate tsunami and inundation waves, dam breaches, and others. Ignoring the friction effects, the SWE system can be writtenas a system of hyperbolic conservation laws with a source term describing the effects of the varying water bed or bathymetry;this source term vanishes in the case of a at bathymetry. Recently, the research on numerical methods for the shallow waterequations became popular for two main reasons or challenges: The SWE system features equilibrium states/stationary solu-tions in which the nonzero ux divergence is exactly balanced by the source term. In their native form, usually most numerical

schemes fail to generate equilibrium state solutions and they generate nonphysical waves, oscillations, and instabilities becausethe ux divergence and the source term lack well-balancing in their discretization. Well-balanced schemes were developed in[12,17–26] in a way to satisfy the steady state requirement such as the lake at rest problem. Another important feature in thesimulation of the SWE problems is the appearance of wet and dry areas that are due to the initial conditions or to the ow of water in the computational domain. Here again most numerical schemes fail to handle the interaction between wet and dryzones and generate negative water heights and other instabilities. The development of well-balanced schemes with wetting anddrying capabilities formed another challenge among the numerical community, and several numerical schemes were recentlydeveloped [27–30] to fulll both the lake at rest and the wetting and drying constraints.

In this paper we develop, analyze and implement a new unstaggered, well-balanced, non-oscillatory, and second-order accu-rate central scheme for the one-dimensional system of shallow water equations on irregular bathymetry with wet and dry zones.The lake at rest constraint will be exactly satised at the discrete level by following the surface gradient method developed in[12,26] which discretizes the water height according to the discretizations of the water level and the bottom topography func-tions. On the other hand wet and dry zones will be carefully treated by introducing a new technique that corrects the slopes of

the linearized water height function over the control cells in the forward and backward projections steps in a way to ensure bothwater conservation and non-negative water height values. We show that the resulting scheme is a well-balanced scheme thatexactly maintains the steady state requirement at the discrete level when lake at rest problems are considered and also allowsproper wave run-ups and withdraws on coastal slopes and shorelines.

The rest of the paper is organized as follows:Section 2is dedicatedto reviewing the shallow water equations, their properties,and their equilibrium states and constraints. In Section 3 we develop the well-balanced central scheme for the shallow waterequations with wet and dry states; we describe the well-balancing technique that ensures the lake at rest constraint and thewet/dry treatment that allows a proper propagation of water waves on shores and islands. Section 4 is devoted to the validationand application of the developed scheme; we perform several numerical experiments from the recent literature, and we conrmthe robustness and potential of the proposed scheme. We end the paper with some concluding remarks and perspectives forfuture work in Section 5.

2. Shallow water equations

Shallow water equations are commonly used to mathematically model rapidly varying free surface ows such as dambreaches, oods and inundation waves, tidal waves in oceans and lakes and many others. The system of shallow water equa-tions is derived from the conservation principles of the mass and momentum, and under the main assumption of hydrostaticpressure distribution, the SWE system is a time dependent two-dimensional system of hyperbolic balance laws. The conserva-tive one-dimensional version of the SWE system reduces to

∂t u + ∂ x f (u ) = S (u , x) , t > 0, x ∈ ⊂ R ,u ( x, 0) = u 0( x) ,

(1)

with x and t are the spatial and temporal independent variables, respectively. The computational domain is an interval of the real axis. Furthermore, the unknown vector solution u ( x, t ) , the ux function f (u ) , and the source term S (u , x) are given asfollows:

u ( x, t ) =h

hv , f (u ) =

hv

hv 2 + 12 gh2 , and S (u , x) =

0

− gh dz dx

. (2)

Here h( x, t ) denotes the water height, v( x, t ) is the velocity in the x-direction, g is the gravitational constant, and z ( x) denotes thebottom topography function (Fig. 1). When the waterbed is at, i.e. z ( x) = constant, the right hand side of system (1) vanishesand the resulting system reduces to a homogeneous hyperbolic system with real eigenvalues λ 1 = v − gh and λ 2 = v + ghand linearly independent eigenvectors.

The SWE system features equilibrium solutions that are solutions to the system∂ x(hv ) = 0,

∂ x(hv 2 + gh2/ 2) = − dhdz dx

.(3)

One particular equilibrium state that we would like to satisfy in this work is the lake at rest equilibrium state dened byv = 0,h + z = constant. (4)


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R. Touma/ Applied Mathematical Modelling 000 (2015) 1–17 3


0 1 2 3 4 5 6 7 8 9 10 110

1

2

3

4

5

6

h(x,t)

z(x)

Fig. 1. Free surface ow over variable bottom topography, featuring a lake at rest state, wetting/drying, and a wave run up.

Numerical methods for the shallow water equations usually fail to satisfy the lake at rest equilibrium state unless a proper well-balanced discretization of the source term is performed. Different schemes were recently [11,12,17,23,26,28,31]to satisfy the lakeat rest equilibrium state.

Another important feature that we would like to properly handle in this work is the ow of water on a wet and dry areainside the computational domain; this usually arises while run-up of waves on shorelines as is illustrated in Fig. 1. Numericalschemes, in their native forms, usually fail to properly handle the wet/dry states and they generate negative water heights beforethey break out. In this work we aim to develop a new well-balanced unstaggered central scheme capable of wetting and drying(WB-UCS-WD) that preserves the water height positivity constraint in the neighborhood of wet/dry states.

3. Well-balanced central schemes for the SWE systems with wet and dry states

In this section, we develop a new well-balanced central scheme for the shallow water equations on variable bottom topogra-phies with wet and dry states. The proposed central scheme follows a classical nite volumes method construction. We start bypartitioning the computational domain = [a , b] ⊂ R using control cellsC i = [ xi− 1/ 2, xi+ 1/ 2] of equal width x = xi+ 1/ 2 − xi− 1/ 2.The nodes xi, centers of the cells C i, are obtained by setting xi = a + i x for i ∈ N . Similarly we dene the dual cells Di+ 1/ 2 =[ xi , xi+ 1] centered at the nodes xi+ 1/ 2 = xi + x/ 2. The time step will be denoted by t , and for n ∈N we set and t n = n t .

Next, for a scalar function u( x, t ), we will use the notation uni to denote the numerical estimate of u( xi, t n), and for any scalarfunction g ( x) admitting point valueswe denote by g i the numerical approximateof g ( xi). Furthermore we will denote the discretespatial forward, backward and centered differences approximating g x( xi) as follows:

D+ ( g i) = g i+ 1 − g i

x ,

D− ( g i) = g i − g i− 1

x ,

D0( g i) = g i+ 1 − g i− 1

2 x ,

with g i ± 1 approximates g ( xi ± x). Furthermore we introduce the average and jump notations, respectively as follows:

uni+ 1/ 2 =uni + u

ni+ 1

2 , uni =

uni− 1/ 2 + uni+ 1/ 2

2 ,

[[u]]i+ 1/ 2 = ui+ 1 − u i, [[u]]i = ui+ 1/ 2 − u i− 1/ 2.In a similar way we will use the notations

[[u]]si = usi+ 1/ 2 − usi− 1/ 2, [[(us) ]]i = usi+ 1/ 2 − usi− 1/ 2 .

3.1. Well-balanced central nite volume method for the SWE system

Without any loss of generality, we assume that the numerical solution u ni to system (1) is known at time t n at the centers

xi of the control cells C i. In order to compute un+ 1i at time t

n+ 1= t

n+ t we will follow a classical nite volume approach thatevolves a piecewise linear numerical solutionL i( x, t n ) dened at the nodes xi, and that approximates the solution u( x, t n ) on C i.



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L i( x, t n ) is given by

L i( x, t n ) = u ni + ( x − xi)( uni ) , ∀ x ∈ C i. (5)

(u ni ) is a limited numerical gradient that approximates the spatial partial derivative ∂∂ xu ( x, t

n ) | x= xi . In this work, we use vanLeer’s monotonized centered limiter (MC− θ ), where the slope of the interpolant is calculated componentwise as follows: thekthcomponent (unk, i) of (u

ni )

(unk, i) = minmod θ [[unk]]i− 12

x ,

unk, i+ 1 − unk, i− 12 x , θ

[[unk]]i+ 12 x

;

here 1 ≤ θ ≤ 2, and the minmod function is dened by

minmod(a , b, c ) =sign(a) min{|a | , | b| , | c |} , if a , b, c have the same sign,0, otherwise.

To construct the numerical solution u n+ 1i at time t n+ 1 we start by integrating the hyperbolic balance law in system (1) on the

rectangle Rni+ 1/ 2 = [ xi, xi+ 1] × [t n , t n+ 1] and then we apply Green’s theorem to the integral to the left; we obtain

∂Rn

i+ 12

( f (u )dt − u dx) =

t n+ 1

t n

xi+ 1

xi

S (u )dxdt

Expanding the integral to the left over the four sides of the rectangle Rni+ 12

, we obtain

− xi+ 1 xi u ( x, t n) dx + t n+ 1

t n f (u ( xi+ 1, t ))dt + xi+ 1 xi u ( x, t n+ 1) dx

− t n+ 1t n f (u ( xi, t ))dt = t n+ 1t n xi+ 1 xi S (u ) dxdt . (6)We assume that at time t n+ 1 and for x ∈ Di+ 1/ 2 the solution u ( x, t n+ 1) ≈ L i+ 1/ 2( x, t n+ 1) is a piecewise linear function dened atthe center xi+ 1/ 2 of the cells Di+ 1/ 2; the Mean-Value theorem gives

xi+ 1

xiu ( x, t n+ 1)dx = xL

i+ 1/ 2( x

i+12

, t n+ 1) = xu n+ 1i+

12

.

In a similar way, since at time t n and for x ∈ C i, the solution u ( x, t n ) ≈ L i( x, t n ) is a piecewise linear function, the Mean-Valuetheorem leads to

xi+ 1 xi u ( x, t n )dx = xi+ 1/ 2 xi u ( x, t n)dx + xi+ 1 xi+ 1/ 2 u ( x, t n )dx=

x2

L i( xi+ 14 , t n) +

x2

L i+ 1( xi+ 34 , t n) := xu ni+ 12 . (7)

Eq. (7) denes the forward projection step that interpolates the numerical solution at time t n as follows:

u ni+ 12 =12

L i xi + x

4 , t n + L i+ 1 xi+ 1 −

x4 , t

n

= uni+ 12 − x

8 [[(un) ]]i+ 12 . (8)

Eq. (6) can now be rewritten as

u n+ 1i+ 12

= u ni+ 12 −1 x t n+ 1t n f (u ( xi+ 1, t ))dt − t n+ 1t n f (u ( xi , t ))dt

+1 x t n+ 1t n xi+ 1 xi S (u )dxdt . (9)

As for the ux integrals in Eq. (9), they are approximated with second-order of accuracy using the midpoint quadrature rule asfollows:

t n+ 1

t n f (u ( xi , t ))dt ≈ t f (u n

+ 12i ), (10)



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where the predicted solution values at time t n+ 1/ 2 are obtained using a rst-order Taylor expansion in time as well as the balancelaw (1) as follows:

u ( xi, t n+12 ) ≈ u ( xi , t n) +

t 2 u t ( xi, t

n)

≈ u ni +t

2 − f (u ) x| ( xi ,t n ) + S (u ) | ( xi ,t n )

≈ u ni +t

2 − ( f ni ) + S ni := un+ 1

2i ,

(11)

with ( f ni ) approximates the ux divergence and S ni ≈ S ( xi, u

ni ) approximates the source term at time t

n on the cell C i withsecond-order of accuracy and is dened with the aid of a sensor function to ensure well-balancing as follows:

S ni = S ni,L + S

ni,R + S

ni,C , (12)

with

S ni,L =s2i (1 − si) ( 2 − si)

6 0

− ghni θ D− ( z i),

S ni,R =s2i (1 + si) ( 2 − si)

2 0

− ghni θ D+ ( z i),

S ni,C = si (si + 1) ( si − 1)6 0− ghni D0( z i)

.

(13)

Here 1 ≤ θ ≤ 2 is the MC− θ parameter used in the gradients limiting in the forward projection step as well as in the predictorstep.

The sensor function si, appearing in the discretization of the source term, forces the discretization of the source term (13) tofollow the same discretization of the term h x appearing in the ux divergence and is dened by

si =

− 1 if hi = θ D− hi,1 if hi = θ D+ hi,0 if hi = 0,2 if hi = D0hi.

(14)

Onthe other hand, the integralof the source terminEq. (9)is also approximated with second-orderof accuracy using centereddifferences and the midpoint quadrature rule as follows:

t n+ 1t n xi+ 1 xi S (u ( x, t )) )dxdt ≈ t x S (u n+ 12i , u n+ 12i+ 1 )with

S (u n+12

i , un+ 12i+ 1 ) =

0− ghn+ 1/ 2

i+ 12D+ ( z i)

. (15)

Eq. (9) combined with Eqs. (10) and (15) reduces to

u n+ 1i+ 12

= u ni+ 1/ 2 − tD+ f (un+ 12i ) + tS u

n+ 1/ 2i , u

n+ 1/ 2i+ 1 . (16)

From Eq. (16), we see that the numerical solution u n+ 1i+ 12 , computed at time t n+ 1, is obtained at the center of the control cells

Di+ 1/ 2. A backward projection step is therefore required to generate the numerical solution u n+ 1i on the cells C i as follows:

u n+ 1i = un+ 1i −

18 (u

n+ 1i+ 1/ 2) − u

n+ 1i− 1/ 2) = u

n+ 1i −

x8 [[(u

n+ 1) ]]i, (17)

where (u n+ 1i+ 1/ 2) denotes a limited numerical gradient that approximates the spatial partial derivative ∂∂ x u ( x, t

n+ 1) | x= xi+ 12

. Thiscompletes the description of the central nite volume scheme. Next we show that the proposed nite volume scheme (16)preserves the lake at rest constraint of the SWE system (1); we rst show that un+ 1i+ 1/ 2 = u

ni+ 1/ 2. In that context, we have the

following theorem:

Theorem 3.1. Assume that the numerical solution u ni of the one-dimensional SWE system is generated using the nite volume method(16), and assume further that u ni satises the lake at rest constraint (3) at the discrete level at time t

n , i.e.,

v ni = 0, and hni + z i = constant , ∀i, (18)



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then,

(a) The predicted numerical solution obtained using Eq. (11) is such that un+ 1/ 2i = uni .

(b) The computed solution at time t n+ 1 on the dual cell D i+ 1/ 2 obtained using Eq. (16) and the forward projection solution obtainedusing Eq. (15) are equal, i.e., un+ 1i+ 1/ 2 = u

ni+ 1/ 2.

Proof 3.1. First we establish (a). If for example in the prediction step (11) h x( xi, t n ) is discretized using the forward difference,i.e., h x( xi, t n ) ≈ (hni ) = θ D+ h

ni , then the sensor function si takes on the value 1, and the discretized source term reduces to

S ni = S ni,R. In addition, since u ni satises the steady state requirement at time t n , then v ni = 0 and f (u ni ) = f ni = 0, 12 g (hni )2 T so( f ni ) = 0, ghni (hni )

T .Therefore the prediction step becomes

u n+12

i = uni +

t 2

0− ghni (h

ni )

+ 0

− ghni θ D+ ( z i)

= u ni +t

2 0

− ghni [(hni ) + θ D+ ( z i)]

= u ni +t

2 x 0

− ghni [θ(hni+ 1 + z i+ 1) − θ(hni + z i)].

But since hni + z i = H = constant for all i, then θ (hni+ 1 + z i+ 1) − θ(hni + z i) = 0leading to

u n+1 2

i = uni .

A similar proof can be done for the other values of the sensor function sni .Next we establish (b), i.e., we show that u n+ 1

i+ 12= u n

i+ 12. Since the numerical solution u ni satises the lake at rest equilibrium

state (4) at time t n at the discrete level, i.e., Eq. (18), and taking into account part (a) of Theorem 3.1, then we get,

f (u n+12

i+ 1 ) = 0

12 g (h

n+ 12i+ 1 )

2 and f (un+ 12i ) =

012 g (h

n+ 12i )

2 .

Substituting in Eq. (16), we get

un+ 1i+ 12 = u

ni+ 12 −

t x

0

12 g (hn+ 12i+ 1 )2 −

0

12 g (hn+ 12i )2

+ t 0

− ghn+12

i+ 1/ 2D+ ( z i).

Performing basic algebra operations, we obtain

u n+ 1i+ 12

= u ni+ 12 −t x

0

hn+12

i+ 1/ 2 (hn+ 12i+ 1 + z i+ 1) − (h

n+ 12i + z i)

.

But since un+12

i = uni and h

ni + z i = H = constant, for all i, then (h

ni + z i) − (h

ni+ 1 + z i+ 1) = 0. Therefore, (h

n+ 12i + z i) − (h

n+ 12i+ 1 +

z i+ 1) = (hni + z i) − (hni+ 1 + z i+ 1) = 0, leading to u

n+ 1i+ 1

2

= u ni+ 1

2

.

Remark 3.1. Theorem 3.1 ensures that if the numerical solution u ni satises the steady state equilibrium state (4) at the discretelevel at time t n on the nodes xi, then the computed solution u n+ 1i+ 1/ 2 at time t

n+ 1 will satisfy the same equilibrium state, but onlyon the nodes xi+ 1/ 2. This does not guarantee that the back-projected numerical solution u n+ 1i obtained using Eq. (11) will satisfythe equilibrium state (4) at the discrete level. Therefore an additional treatment is required. In this work, we follow the surface gradient method suggested by Zhou et al. [12,16,26], for SWE systems on completely ooded domains to the case of systems of SWE systems with wet and dry regions.

Following the surface gradient method, we calculate the numerical derivative of the water height h( x, t ) component in termsof the water level function H ( x, t ) = h( x, t ) + z ( x) . To do so, we assume that the bottom topography function values are knownat the cell-interfaces xi+ 1/ 2, i.e., the values z i+ 12 are given at the nodes xi+ 12 . At the cell centers xi, we set z i =

12 ( z i+ 12

+ z i− 12). The

waterbed function z ( x) is reconstructed on the control cells C i using linear interpolants as follows:

z ( x) = z i +1 x( z i+

12 − z i− 12 )( x − xi) ∀ x ∈C i.



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Next, we linearize the water height function h( x) on the control cells C i using the linearization of the water level function H ( x).For the linearization of the water level function H ( x, t n ) ≈ H ni + (H

ni ) ( x − xi) on the cells C i we limit the numerical derivatives of

H ni = hni + z i; the numerical spatial derivative of the water height function is then obtained as follows:

(hni ) = (H ni ) −

1 x

( z i+ 1/ 2 − z i− 1/ 2) . (19)

Eq. (19) is used in the forward projection step to calculate the rst component of u ni+ 1/ 2 on the dual cells in Eq. (8).

Similarly for the back projection step of the water height component in un+ 1i+ 1/ 2 on the original cells C i (i.e., Eq. (17)), we rstdene the water level values H̃ n+ 1i+ 1/ 2 at the centers of the dual cells as follows:

H̃ n+ 1i+ 1/ 2 = hn+ 1i+ 1/ 2 + ˜ z i+ 1/ 2, (20)

where ˜ z i+ 1/ 2 = z i+ 1/ 2 −12 z i+ 1/ 2 − z i+ 1/ 2 is the corrected bottom elevation due to the fact that the function z ( x) is linear only on

the cells C i and not on the staggered cells Di+ 1/ 2.Next, we compute the limited numerical derivative of the water height function (hn+ 1i+ 1/ 2) using the water level values H̃ i+ 1/ 2

obtained over the corrected waterbed as follows:

(hn+ 1i+ 1/ 2) = (H̃ n+ 1i+ 1/ 2) −

1 x

( z i+ 1 − z i) . (21)

This discretization of the spatial derivative of h( x, t ) will ensure that the projected numerical solutionu n+ 1i back onto the originalcells C i will satisfy equilibrium state requirement in the case of a the lake at rest problem as is stated in Theorem 3.2. Theorem 3.2. Assume that the numerical solution u n+ 1i of the system of shallow water Eq. (1) is computed using the nite volumemethod (16) under the hypotheses of Theorem 3.1, and following the surface gradient method for the forward projection step (8), (19)and the backward projection step (17), (20), (21) for the water height component, then the proposed nite volume scheme is a well-balanced scheme that exactly preserves the lake at rest requirement at the discrete level (4)and therefore the equation u n+ 1i = u

ni holds

for all i.

Proof 3.2. To show that the lake at rest equilibrium state (18) remains maintained at time t n+ 1 provided it was as such at timet n we will proceed component wise and show that u n+ 1i = u

ni , for all i. Below we present the proof for the h component. We

note that in the lake at rest equilibrium state (4) the hv component does not change in time because v = 0 and because of thewell-balanced discretization.

We use the surface gradient method for both the forward and backward projection steps; for the forward projection we have

hni+ 1/ 2 = hni+ 1/ 2 +

x8 (h

ni ) − (h

ni+ 1) , (22)

where the derivatives (hni ) are discretized using the water level function H ni = h

ni + z i as described in Eq. (19). Substituting in

Eq. (22) one obtains (while taking into account that H ni is constant for all i)

hni+ 1/ 2 = hni+ 1/ 2 −

12 z i+ 1/ 2 − z i+ 1/ 2 . (23)

Similarly, for the backward projection step we apply the surface gradient method and one obtains (while taking into accountthat H

n+ 1/ 2i+ 1/ 2 = h

n+ 1/ 2i+ 1/ 2 + z i+ 1/ 2 is constant for all i)

hn+ 1i = hn+ 1i +

18 − [[ z ]]i− 1/ 2 + [[ z ]]i+ 1/ 2 . (24)

Substituting Eq. (23) in (24) leads to hn+ 1i

= h

ni . Therefore we conclude that if the steady state (18) was satised at time t

n

, thenit will remain as such at the next timet n+ 1.

3.2. Treatment of wet/dry states

When shallow water equation problems with wet and dry regions are considered, numerical instabilities usually arise unlessa proper care is taken while computing the numerical solution. In the case of unstaggered central schemes, the way the forwardand backward projection steps are performed is crucial for ensuring a physically admissible numerical solution. More precisely,in the neighborhood of wet and dry regions, the appearance of spurious oscillations and other numerical instabilities is mainlydue to the gradient limiters that yield an interpolated numerical solution falling below the waterbed on the dual cells in theforward projection step, or an interpolated water height lying below the water bed function on the original cells in the backwardprojection step; in either case, the generatednumerical solution becomesphysically non-admissible, and therefore an additionaltreatment is required. In this work we propose a new wetting/drying treatment procedure for the SWE systems that allows clean

forward and backward projection steps without yielding negative water heights, and conserves the total amount of water acrossthecomputational domain. Theproposed wetting/dryingprocedure is triggeredwhen an interpolation hadyield a negative water



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x

y

x i

x i + 1

z + η n i

z + η ∗ ni

z +

η n

i + 1

x i + 1 / 2

x i + 1 / 4

x i + 3 / 2

x i

− 1 / 2

Fig. 2. The corrected forward projectionstepavoids negative water height valueshni+ 1/ 2 and conservesthe cell’s wateraveragevaluehni in C i with hni+ 1 = 0 inC i+ 1.

x

y

x i

x i + 1

z + η

n i

z +

η ∗ n

i + 1

z +

η n

i + 1

x i + 1 / 2

x i + 3 / 4

x i + 3 / 2

x i

− 1 / 2

Fig. 3. Correcting the forward projection step leading to hni+ 1/ 2 = 0 while conserving the cell’s water average value hn1 in the cell C i with hni+ 1 = 0 in cellC i+ 1.

height value, then we redene the slope of thewaterheight interpolant appropriately on theneighboring control cells in a way to

ensure a non-negative water height interpolated value, which guarantees a physically admissible numerical solution. To betterdescribe the effects of wet and dry states on the interpolated water height values, we consider the following case where thenumerical solution u ni = (hni , hni uni ) is such that hnk > 0 for k ∈ {i − 1, i} and h

ni+ 1 = 0.

Performing a forward projection step such as Eq. (8) may lead to a negative water height value (hni+ 1/ 2) in uni+ 1/ 2 if the slope of

the interpolant ηni ( x) that approximates h( x, t n ) for x∈C i is too steep as is illustrated in Fig. 2. The water height interpolantηni ( x)

on the cell C i at time t n is dened by

ηni ( x) = hni + ( x − xi)( h

ni ) , x ∈C i. (25)

Note that ηni ( x) ≈ h( x, t n ) for x ∈ C i and ηni ( xi) = h

ni > 0 on a wet cell. For a dry cell C i (i.e., if h

ni = 0), we set η

ni ( x) = 0∀ x ∈C i. A

negative water height hni+ 1/ 2 in the interpolated solution arises for example when ηni ( x) is too steep so that η

ni ( xi+ 1/ 4) < 0 and

ηni+ 1( xi+ 3/ 4) < | ηni ( xi+ 1/ 4) | . This results inh

ni+ 1/ 2 = 0.5 η

ni ( xi+ 1/ 4) + η

ni+ 1( xi+ 3/ 4) < 0.A wetcellC i neighbor to a dry or nearly dry

cellC i+ 1, may result in such a situation. This is illustrated inFig. 2where we show thewater level function z + ηni over the wet cell

C i and the water level function z + ηni+ 1 over the dry cellC i+ 1. Fig. 2 shows also the corrected water level function z + η

∗ni (dottedline) over the wet cell C i that ensures hni+ 1/ 2 = 0 on the dual cell Di+ 1/ 2. Fig. 3 shows a similar case where the forward projection



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step leads to a negative water height value hni+ 1/ 2 whenever hni = 0 and h

ni+ 1 > 0; the corrected water height interpolant η

∗ni+ 1( x)

for x ∈C i+ 1 will guarantee non-negative values of hni+ 1/ 2.The treatment we propose in this work is as follows. Without any loss of generality, we can assume that the water height

component hni in the numerical solutionuni at time t

n is positive or zero. Our challenge is to generate the numerical solutionu n+ 1iin a way the water heighthn+ 1i remains non-negative. Explicitly,we need to make sure that theforward projection step leading tou ni+ 1/ 2 and the backward projection step leading to u

n+ 1i fulll the non-negativity requirement of the water height. We propose

the following modications to the forward and backward projection steps. If in the forward projection step, the water heighthni+ 1/ 2 in the cellDi+ 1/ 2 becomesnegative, then we sethni+ 1/ 2 to be zero and we correct the slopes of the water height interpolantsin the cells C i and C i+ 1 to ensure water conservation across the computational domain. Two cases might arise: If hni ≥ (for somepreassigned tolerance = 10− 9) we re-linearize the water height function on cell C i by taking h x( xi, t n ) ≈ −

hni x/ 4 , i.e., we set

(hni ) = − hni

x/ 4 , otherwise if hni < , weset hni = 0 andh x( xi , t

n ) = (hni ) = 0. Similarly, if hni+ 1 ≥ we re-linearize thewaterheight

function h( x, t n) on the cellC i+ 1 and we take h x( xi+ 1, t n ) ≈hni+ 1 x/ 4 , otherwise we set hni+ 1 = 0 and h x( xi+ 1, t

n ) = 0. Furthermore, tomaintain well balanced property of the developed scheme, the discretization of z x in the source term in Eqs. (1) and (2) shouldmimic the discretization of h x. We therefore extend denition of the sensor si in Eq. (14) as well as the discretization of thesource term S ni in Eq. (13) to include the following two cases, whenever a negative water height value hni+ 1/ 2 had been detected,as follows.

If hni > , then in addition to setting (hni ) = −

hni x/ 4 , weset z x( xi) ≈

z i+ 1/ 2− z i( x/ 2) for thediscretization of z x( xi) in theapproximation

of the source term. Similarly if hni+ 1 > , then in addition to setting(h

ni+ 1) =

hni+ 1 x/ 4 , we set z x( xi) ≈

z i+ 1− z i+ 1/ 2( x/ 2) . A similar treatment

is to be applied for the backward projection step that generates hn+ 1i . This will ensure both a non-negative forward projectedwater height value hni+ 1/ 2 and a backward projected water height value h

n+ 1i , while maintaining the well-balanced discretization

of the source term according to the discretization of the ux divergence. We nally note that the proposed treatment of wet drystates ensures a physically admissible numerical solution with non-negative water height while maintaining the well-balancedproperty of the numerical base scheme, but it does not necessarily ensure the lake at rest constraint along with wet/dry interac-tion constraint, simultaneously. In fact, and as is discussed in [32] it is not easy to make the scheme still satisfy the lake at restconstraint after a wetting/drying treatment.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6tf =10

100 grid pointsExact solution

Fig. 4. Lake at rest problem: prole of the water level at time t = 10.

Table 1Lake at rest problem: experimental order of convergence measured in the L1-norm.

N L1 error ρ Order

50 1.96E− 02100 3.73E− 03 2.39200 8.28E− 04 2.17



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−15 −10 −5 0 5 10 15−1

0

1

2

3

4

5

6

α = 0

x

Water level and velocity

0 0.5 1 1.5 20

5

10

15

α = 0

time t

Front position

WaterbedWater levelVelocity

Exact solutionNumerical solution

(a) α = 0

−15 −10 −5 0 5 10 15−1

0

1

2

3

4

5

6

α = π

/ 6 0

x


0 0.5 1 1.5 20

5

10

15

α = π

/ 6 0

time t

Front position



(b) α = π/ 60

−15 −10 −5 0 5 10 15−1

0

1

2

3

4

5

6

α = − π

/ 6 0

x



0 0.5 1 1.5 20

5

10

15

α = − π

/ 6 0

time t

Front position


(c) α = − π/ 60

Fig. 5. Dam break over an inclined plane.



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4. Numerical experiments

In this section we apply the developed well-balanced numerical scheme and solve classical one-dimensional shallow waterequation problems with at or variable bottom topography functions.

4.1. Lake at rest problem with smooth bottom topography

The rst experiment features a lake at rest problem with a smooth bottom topography. This experiment is meant to validatethe well-balanced property of the proposed scheme and its second-order of accuracy. The bottom topography is dened by thefunction z = sin2 (π x) over then interval [0, 1]. The initial water level is set to be H ( x, 0) = h( x, 0) + z ( x) = 5 and the initialvelocity is v ( x, 0) = 0. The numerical solution is computed at the nal time t f = 10 on 100 grid points and is compared to theexact solution (lake at rest solution). The obtained numerical results are reported in Fig. 4, where we compare the water levelobtained on 100 grid points (◦ curve) to the exact solution of the problem (solid curve). Both curves are in perfect match thusconrming the well-balanced property of the proposed scheme.

The L1 error and the order of convergence of the numerical solution on an increasing mesh was computed; the obtainedresults, reported in Table 1, validate the order of accuracy of the numerical scheme.

4.1.1. Dam break over a planeThis experiment features a dam break over inclined planes with various angles of inclination as suggested in [28,31,33]. The

computational domain is the interval [− 15, 15] which we discretize using 200 grid points. The bottom topography function isdened by the function z ( x) = x tan α , where α denotes the inclination angle. In this example we consider three inclinationsα = − π / 60 (downhill inclination), α = 0 (at), and α = π / 60 (uphill inclination). The initial conditions for this test case are asfollows:

u( x, 0) = 0, h( x, 0) =1 − z ( x) , x < 0,0, otherwise.

We apply the proposed scheme and compute the numerical solution at timet = 2. The obtained results are reported in Fig. 5(a)–(c), for α = − π / 60, α = 0, and α = π / 60, respectively. The left column shows the water elevation and velocity along the wa-terbed at the nal time, while the right column shows the front location at each time t ∈ [0, 2] obtained using the proposednumerical scheme and compared to the exact analytical front position given by x f (t ) = 2t g cos(α) − 0.5 gt 2 tan (α) ; the nu-merical front location is dened to be the rst cell-center (from left to right) where the water height exceeds the value = 10− 9.The obtained numerical results are in perfect match with their corresponding analytical solution and in perfect agreement with

the results reported in [28,33], thus conrmingthe potential of the proposed schemeto handle shallow waterequation problemson variable topographies with wet and dry states. The water mass m(t ) at each time step t on the computational domain [− 15,15]is computed and compared to the initial water mass m0. The obtained results are reported in Fig. 6 where we show the graph of the curve m(t )/m0 on 50, 100 and 200 grid points. The obtained results conrm the conservation of the water mass as x tendsto zero.

4.1.2. Parabolic bowlNow we consider the parabolic bowl problem as considered in [33]. This problem was previously considered in the context of

shallow water equations with friction term in [33]. The computational domain is the interval [− 5000, 5000] and the water bed

0 0.5 1 1.5 2 2.50.9986

0.9988

0.999

0.9992

0.9994

0.9996

0.9998

1

t

m ( t ) / m 0

50 points100 points200 points

0 0.5 1 1.5 2 2.51

1.0001

1.0002

1.0003

1.0004

1.0005

1.0006

t

m ( t ) / m 0

50 points100 points

200 points

Fig. 6. Dam break over an inclined plane: graph of the curve m(t )/m0 on the interval 0≤ t ≤ 2 obtained on 50, 100, and 200 spatial grid points.



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is the parabolic bowl dened by z ( x) = h0( x/ a)2, where h0 = 10 and a = 3000 are two constants. The initial water height h( x) isgiven by the equation

h( x, t = 0) = h0 −B2

2 g −Bx2a 8h0 g − z ( x) ,

where B = 5 is a constant. The initial velocity v ( x, t = 0) is set to zero. The exact solution of this problem at any time t is given by

the equation

h( x, t ) = h0 −B2

4 g cos(2ωt ) −B2

4 g −Bx2a 8h0 g cos(2ωt ) − z ( x) ,

−5 00 0 −4 00 0 −3 00 0 −2 00 0 −1 00 0 0 1 00 0 20 00 3 00 0 40 00 5 00 00

5

10

15

20

25

30

x

W

a t e r e l e v a t

i o n

WaterbedNumerical SolutionExact solution

(a) Water height at t = 1000

− 50 00 − 40 00 − 30 00 − 20 00 − 10 00 0 10 00 20 00 30 00 4 000 50 000

5

10

15

20

25

30

x

W

a t e r e l e v a t

i o n


(b) Water height at t = 2000

−5 00 0 −4 00 0 −3 00 0 −2 00 0 −1 00 0 0 1 00 0 20 00 3 00 0 40 00 5 00 00

5

10

15

20

25

30

x

W a t e r e l e v a t

i o n


(c) Water height at t = 3000

− 50 00 − 40 00 − 30 00 − 20 00 − 10 00 0 10 00 20 00 30 00 4 000 50 000

5

10

15

20

25

30

x


i o n


(d) Water height at t = 4000

− 50 00 − 40 00 − 30 00 − 20 00 − 10 00 0 10 00 2 000 30 00 4 00 0 5 0000

5

10

15

20

25

30

x


i o n


(e) Water height at t = 5000

−5 00 0 −4 00 0 −3 00 0 −2 00 0 −1 00 0 0 1 00 0 2 00 0 3 00 0 4 00 0 5 00 00

5

10

15

20

25

30

x


i o n

WaterbedNumerical Solution

Exact solution

(f) Water height at t = 6000

Fig. 7. Parabolic bowl problem.



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and the exact location of the water is located between the points x1 = − a − Bωa2

2 gh0cos(ω t ) and x2 = a − Bωa

22 gh0

cos(ω t ) , whereω = 2 gh0/ a. The computational domain is partitioned using 200 control cells and the numerical solution is calculated usingthe proposed WB-UCS-WD scheme at the nal time t = 6000. The obtained numerical results are reported in Fig. 7 where weshow the water height at different times. The exact solution at each time is also shown in Fig. 7, where we see a perfect matchbetween our generated results and the exact reference solution.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4

t=0


−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4

t=0.07

(b) Water height at t = 0 .07

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4

t=0.148

(c) Water height at t = 0 .148

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4

t=0.278

(d) Water height at t = 0 .278

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4

t=0.49

(e) Water height at t = 0 .49

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4

t=0.8

(f) Water height at t = 0 .8

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4

t=0.148

(g) Water height at t = 1 .25

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4

t=1.5

(h) Water height at t = 1 .5

Fig. 8. Symmetric double dam-break problem.



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0 1 2 3 4 5 6

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

t

m

( t ) / m

0

50 gridpoints100 gridpoints200 gridpoints

Fig. 9. Symmetric double dam-break problem: graph of the curve m(t )/m0 on the interval [0,6] obtained on 50, 100, and 200 grid points.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t=0


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t=0.1


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t=0.19


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t=0.3

(d) Water height at t = 0 .3

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t=0.42

(e) Water height at t = 0 .42

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t=0.49

(f) Water height at t = 0 .49

Fig. 10. Dam break problem with irregular bumps.



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4.1.3. Two symmetric dam break problemsWe consider now a symmetric double dam-break problem facing a steep triangular bump. The computational domain is the

interval [− 2, 2] and reective boundary conditions are set at the endpoints of the domain. The variable waterbed function z ( x)and the initial water height function h( x, 0) are given by

z ( x) =

1 if x < − 1,2 + x if − 1 ≤ x ≤ 0,2 − x if 0 ≤ x ≤ 1,1 if x ≥ 1

and h( x, 0) =2 if x < − 1,0 if − 1 ≤ x ≤ 1,

3 if x > 1.The initial water velocity v( x, 0) is set to zero. We discretize the computational domain using 100 grid points and compute thenumerical solution at thenal timet = 1.5 using the MC− (θ = 1.5) limiter. Theobtained results for thewater heightare reportedin Fig. 8(a)–(h). Since the early computations, we see two symmetric rarefaction water waves propagating toward the line x = 0.At the time t = 0.148 the two waves meet in the center of the computational domain and then they reect to form two shockwaves propagating toward the left and right boundaries of the computational domain. By the time t = 1.25 these two shockwaves have reected at the boundaries of the domain and have formed two new rarefaction waves propagating again toward thecenter of the domain. This process is repeated in perfect symmetry about the line x = 0 as is shown in Fig. 8(g) and (h). We havealso computed the water mass m(t ) across the computational domain at each time step and we compared the obtained resultsto the initial water mass m0. Fig. 9 shows the curve m(t )/m0 over the interval [0, t f ] on 50, 100 and 200 grid points. The obtainedresults conrm the conservation of the water mass as x tends to zero.

4.1.4. Dam break problem facing irregular bumpsWe consider for our last experiment a dam break problem facing two irregular bumps with wet and dry states. The compu-tational domain is the interval [− 2, 2.5], and the dam is located at the point x = − 1.5; the initial water height is h( x, 0) = 4 if x < − 1.5 and h( x, 0) = 0 elsewhere. The waterbed features a triangular bump facing the dam and a trapezoidal bump follows

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t=0.67

(a) Water height at t = 0 .67

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t=1.95


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t=3.55


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t=10

(d) Water height at t = 10

Fig. 11. Dam break problem with irregular bumps (continued).



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0 0.5 1 1.5 2 2.5 3 3.50.75

0.8

0.85

0.9

0.95

1

1.0550 gridpoints100 gridpoints200 gridpoints

Fig. 12. Dam break problem with irregular bumps: graph of the curve m(t )/m0 on the interval [0,3.5] obtained on 50, 100, and 200 spatial grid points.

toward the right boundary of the domain as is shown in Fig. 10(a). Reecting boundary conditions are set on both boundaries of the domain. The proposed scheme is applied and the numerical solution is computed at the nal time

t = 10. The obtained re-

sults are reported in Figs. 10(a)–11(d) where we show the water height at different times. As is shown in Fig. 10(a)–(e) the waterfront is moving from left to right crossing the both bumps, then reects at the right boundary of the computational domain, andgoes backward to cross again both bumps before reaching the leftboundary(Figs.10(f)–11(d)). At the timet = 10, wesee that thewater is spread in the regions separated by the two bumps to form three isolated lakes where water waves keep on propagatingback and forth in each of them.

The water mass m(t ) across the computational domain was computed at each time step, and compared to the initial watermass m0. The obtained results are reported in Fig. 12 where we show the graph of the curve m(t )/m0 over the interval [0, t f ] on50, 100 and 200 spatial grid points. The obtained results conrm the conservation of the water mass as x tends to zero.

5. Conclusion

In this work, we developed a new one-dimensional central nite volume method for the numerical solution of systems of

shallow water equations on variable waterbed with wet and dry states. The proposed scheme is a well-balanced unstaggeredcentral nite volume scheme especially designed to satisfy the lake at rest equilibrium state of the shallow water equations andis capable of handling wet and dry states over irregular topographies. The well-balanced component of the scheme is ensuredthanks to the surface gradient method through a properdiscretization of the waterheight in terms of the water level function. Asfor the wetting/drying capability of the scheme, it is ensured through a proper interpolation of the water height components inthe forward and backwardprojection steps, and negative water heights at thewet/dry frontsare avoided by redening the slopesof the interpolants in terms of the water bed elevations at the front locations. The resulting scheme is a second-order schemethat evolves a piecewise linear numerical solution on a single grid, avoids the resolution of the Riemann problems at the cellinterfaces, is well-balanced at thediscrete level dueto a proper linearization of thesourceterm similar to thediscretization of theux terms, preserves the positivity of the water height. The proposed numerical scheme is then validated and classical shallowwater equation problems on variable waterbeds with wet and dry states are successfully solved. The obtained results showvery good agreement with corresponding ones appearing in the recent literature thus conrming the potential and efficiencyof the proposed methods to handle general shallow water equation problems. We are currently working on a two-dimensional

extension of the proposed scheme that allows a physically admissible interaction between wet and dry states in two spacedimensions while preserving the well-balanced property of the numerical base scheme.

References

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Paris 320 (1995) 85–88.[3] A.N. Guarendi, A.J. Chandy, Nonoscillatory central schemes for hyperbolic systems of conservation lawsin three-space dimensions, Sci. World J. 2013 (2013).[4] G.-S. Jiang, E. Tadmor, Nonoscillatory central schemes for multidimensional hyperbolic conservation laws, SIAM J. Sci. Comput. 19 (6) (1998) 1892–1917.[5] R. Touma, P. Arminjon, Central nite volume schemes with constrained transport divergence treatment for three-dimensional ideal MHD, J. Comput. Phys.

212 (2) (2006) 617–636.[6] P. Arminjon, M.-C. Viallon, Convergence of a nite volume extension of the Nessyahu-Tadmor scheme on unstructured grid for a two-dimensional linear

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