Abstract
In this work, we focus on **the** numerical approximation of **the** **shallow** **water** **equations** in two space dimensions. Our aim is to propose **a** **well**-**balanced**, all-regime and positive **scheme**. By **well**-**balanced**, it is meant that **the** **scheme** is able to preserve **the** so-called lake at rest smooth equilibrium solutions. By all-regime, we mean that **the** **scheme** is able to deal with all flow regimes, including **the** low-Froude regime which is known to be challenging when using usual Godunov-type finite volume schemes. At last, **the** **scheme** should be positive which means that **the** **water** height stays positive **for** all time. Our approach is based on **a** **Lagrange**- **projection** decomposition which allows to naturally decouple **the** acoustic and transport terms. Numerical experiments on unstructured meshes illustrate **the** good behaviour of **the** **scheme**.

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31077 Toulouse Cedex 4, France
Abstract
In this communication, we consider **a** numerical **scheme** **for** **the** **shallow**-**water** system. **The** **scheme** under consideration has been proven to preserve **the** positivity of **the** **water** height and to be fully **well**-**balanced**, i.e. to exactly preserve **the** smooth moving steady state solutions of **the** **shallow**-**water** **equations** with **the** topography source term. **The** goal of this work is to prove **a** discrete entropy inequality satised by this **scheme**.

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Abstract
We consider **the** **shallow**-**water** **equations** with Manning friction or topography, as **well** as **a** combination of both these source terms. **The** main purpose of this work concerns **the** derivation of **a** non-negativity preserving and **well**-**balanced** **scheme** that approximates solutions of **the** system and preserves **the** associated steady states, including **the** moving ones. In addition, **the** **scheme** has to deal with vanishing **water** heights and transitions between wet and dry areas. To address such issues, **a** particular attention is paid to **the** study of **the** steady states related to **the** friction source term. Then, **a** Godunov-type **scheme** is obtained by using **a** relevant average of **the** source terms in order to enforce **the** required **well**-balance property. An implicit treatment of both topography and friction source terms is also exhibited to improve **the** **scheme** while dealing with vanishing **water** heights. **A** second-order **well**-**balanced** MUSCL extension is designed, as **well** as an extension **for** **the** two-dimensional case. Numerical experiments are performed in order to highlight **the** properties of **the** **scheme**. Keywords: **shallow**-**water** **equations**, Manning friction, Godunov-type schemes, **well**-**balanced** schemes, moving steady states

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These steady states are highly nonlinear, and exact preservation is **a** challenging task, see **for** instance [ 5 , 41 ] and references within. Note that, **for** **the** case of both source terms, **the** steady states ( 1.9 ) cannot be rewritten under an algebraic form.
In [ 40 , 41 ], **the** authors develop **a** robust numerical **scheme** able to exactly preserve and capture **the** smooth steady states associated with **the** topography and **the** friction source terms. In addition, this **scheme** was proven to be entropy-satisfying in [ 12 ]. We now aim at providing **a** high-order extension in two space dimensions, while retaining **the** robustness property, i.e. **the** preservation of **the** **water** height non-negativity, and **the** essential **well**- balancedness property. First steps have been undertaken in [ 11 ], where **a** **well**-**balanced** second-order MUSCL extension is proposed. Note that other work has been devoted to **the** development of high-order schemes which preserve **the** lake at rest (see **for** instance [ 16 , 22 ]) or **the** moving steady states (see **for** instance [ 43 , 20 , 49 , 19 ]). However, these schemes mostly rely on directly reconstructing **the** algebraic relation ( 1.11 ), which entails **the** added computational cost of having to solve this nonlinear relation **for** h. In addition, **the** friction source term is left mostly untreated. Therefore, our goal is to propose **a** **well**-**balanced** high-order strategy, **for** both friction and topography source terms, that does not rely on solving nonlinear **equations**, and that is applicable to **the** **shallow** **water** **equations** **for** two-dimensional geometries.

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1.3 Purpose and organization
In **the** present work, we are concerned with **the** derivation of **a** numerical **scheme** to approximate **the** weak solutions of ( 1.1 ), which, in addition, accurately captures **the** steady state solutions of ( 1.10 ) and **the** asymptotic regime given by ( 1.7 ). From now on, let us recall that **a** **scheme** able to capture **the** steady states is said **Well**-**Balanced** while **a** **scheme** able to restore **the** asymptotic regimes is said Asymptotic Preserv- ing. **The** **well**-**balanced** schemes were introduced in [ 6 , 18 ] in **the** framework of **the** **shallow**-**water** model with non-flat topography. During **the** two last decades, numerous works were devoted to **the** derivation of **well**-**balanced** schemes (see [ 4 , 16 , 21 , 29 ] **for** **a** non-exhaustive list). More recently, in [ 7 , 27 ], **a** fully **well**-**balanced** Godunov-type **scheme** was introduced. **The** originality of these works stays in **the** incorporation of **the** source term in **the** approximate Riemann solver. This approach allows now to consider extensions to more general systems which include nonlinear source terms. In particular, in [ 28 ], **a** fully **well**-**balanced** **scheme** is derived to approximate **the** weak solutions of ( 1.1 ). Such **a** numerical method exactly captures **the** steady states governed by ( 1.10 ). Concerning **the** derivation of asymptotic preserving schemes, after **the** pioneer work by Jin [ 22 ], several methods were developed to design **a** suitable numerical viscosity in order to restore **the** expected asymptotic diffusive regime (**for** instance, see [ 8 , 9 , 11 – 13 , 17 ]). In [ 10 ], **the** authors introduced **a** technique to morph **the** numerical viscosity into **the** correct diffusive regime. This approach was extended in [ 15 ] in order to deal with **the** system ( 1.1 ) and **the** associated diffusive regime ( 1.7 ).

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Abstract
English version: We are interested in simulating blood flow in arter- ies with variable elasticity with **a** one dimensional model. We present **a** **well**-**balanced** finite volume **scheme** based on **the** recent developments in **shallow** **water** **equations** context. We thus get **a** mass conservative **scheme** which also preserves equilibria of Q = 0. This numerical method is tested on analytical tests.

In this paper we have considered **the** classical 1-D model of flow in an artery, we have presented **a** new numerical **scheme** and some numerical tests. **The** new proposed **scheme** has been obtained in following recent advances in **the** **shallow**-**water** Saint-Venant community. This community has been confronted to spurious flows induced by **the** change of topography in not **well** written schemes. In blood-flows, it corresponds to **the** treatment of **the** terms due to **a** variating initial shape of **the** artery, which arises is stenosis, aneurisms, or taper. To write **the** method, we have insisted on **the** fact that **the** effective conservative variables are: **the** artery area **A** and **the** discharge Q, which was not clearly observed up to now. **The** obtained conservative system has been then discretized in order to use **the** property of equilibrium of ”**the** man at eternal rest” analogous to **the** ”lake at rest” in Saint-Venant **equations**. If this property is not preserved in **the** numerical stencil, spurious currents arise in **the** case of **a** variating vessel. **For** sake of illustration, if **the** terms are treated in **a** too simple way, we have exhibited **a** case with an aneurism which induces **a** non zero flux of blood Q. In **a** pulsatile case, **the** same configuration creates extra waves as **well**. An analogous of dam break (here **a** tourniquet) was used to validate **the** other terms of **the** dicretisation. Other less demanding examples have been performed: linear waves without and with damping in straight tubes. Good behaviour is obtained in those pertinent test cases that explore all **the** parts of **the** **equations**.

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Actually, preliminary numerical simulations show that, even if we start from strictly positive initial data, vacuum may appear in finite time. This occurrence is much more physically relevant with respect to **the** analogous situation in gas dynamics. If vacuum is not expected to appear really in gases, it is fully relevant when dealing with **the** density of cells, i.e.: there are admissible regions without cells, and in some sense it is **the** main goal of **a** biologically consistent model. This is **a** situation somewhat similar to what occurs when dealing with flows in rivers with **shallow** **water** type **equations** [2]. Besides, also looking at **the** numerical approximation, dealing with vacuum needs **for** **a** special care, since we have to guarantee **for** **the** non negativity of **the** solutions [4]. This can be understood at **the** level of **the** associated numerical flux. **A** numerical flux resolves **the** vacuum if **for** all values of **the** approximate solution, it is able to generate nonnegative solutions with **a** finite speed of propagation. In this paper it will be crucial to use schemes with this kind of property.

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January 30, 2012
Abstract
In **the** last decades, more or less complex physically-based hydrologi- cal models, have been developed that solve **the** **shallow** **water** **equations** or their approximations using various numerical methods. Model users may not necessarily know **the** different hypothesis lying behind these develop- ment and simplifications, and it might therefore be difficult to judge if **a** code is **well** adapted to their objectives and test case configurations. This paper aims at comparing **the** predictive abilities of different models and evaluating potential gain by using advanced numerical **scheme** **for** mod- elling runoff. We present four different codes, each one based on either **shallow** **water** or kinematic waves **equations**, and using either finite volume or finite difference method. We compare these four numerical codes on different test cases allowing to emphasize their main strengths and weak- nesses. Results show that, **for** relatively simple configurations, kinematic waves **equations** solved with finite volume method represent an interesting option. Nevertheless, as it appears to be limited in case of **discontinuous** topography or strong spatial heterogeneities, **for** these cases we advise **the** use of **shallow** **water** **equations** solved with **the** finite volume method.

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and u = q = 0) is **a** particular solution to ( 1 ). Since [ 2 ], it is **well** known that **the** topography source term needs **a** special treatment in order to preserve at least this equilibrium. Such schemes are said to be **well**-**balanced** (since [ 9 ]).
In **the** following, we present briefly **the** so-called hydrostatic reconstruction method, which permits, when coupled to **a** positive numerical flux, to obtain **a** family of **well**-**balanced** schemes that can preserve **the** **water** height nonnegativity and deal with dry zones. We show that this method, presented in [ 1 , 4 ] and widely used, fails **for** some combinations of slope, mesh size and **water** height. We give **the** criteria that ensures **the** accuracy of **the** results.

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An equation **for** photons containing **a** frequency drift term modeling **the** Doppler effects and an emission absorption coefficient is considered. On **the** one hand, this frequency drift term involves **a** coefficient κ relative to **the** fluid velocity with which **the** photons interact. This coefficient may vanish, leading to numerical difficulties. On **the** second hand, **the** emission absorption σ is known to be very irregular with respect to **the** frequency, and thus **the** design of **well**-**balanced** **scheme** is **a** mathematical issue. In this work are presented new numerical results **for** **the** spectrally **well**-**balanced** **scheme** in **the** context of highly irregular opacities.

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VARIATIONS
DENYS DUTYKH ∗ AND DIDIER CLAMOND
Abstract. In this study, we propose an improved version of **the** nonlinear **shallow** **water** (or Saint-Venant) **equations**. This new model is designed to take into account **the** effects resulting from **the** large spatial and/or temporal variations of **the** seabed. **The** model is derived from **a** variational principle by choosing **the** appropriate **shallow** **water** ansatz and imposing suitable constraints. Thus, **the** derivation procedure does not explicitly involve any small parameter.

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where n Ω denotes **the** unit vector normal to ∂Ω pointing outward Ω and G : (0, T ) × Γ **A** → R 3
is **a** tangential load term (i.e. G · n Ω = 0). **The** first relation of (1b) is **a** first-order absorbing
boundary condition (ABC) known as **the** Silver-Muller ABC. It is **the** simplest form of ABC **for** Maxwell’s **equations**, and one could alternatively consider higher order ABCs [15] or perfectly matched layers [19]. **The** second equation in (1b) models **the** boundary of **a** perfectly conducting material. Finally, initial conditions are imposed in Ω

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