The proposed strategy can be understood as a natural extension to the present setting of the first- order well-balancedLagrange-Projectionscheme developed in  fortheshallowwaterequations, and of the high order Lagrange-Projectionscheme introduced in  (see also  fora similar approach) forthe barotropic gas dynamics equations. TheLagrange-Projection (or equivalently Lagrange-Remap) decomposition naturally decouples the acoustic and transport terms of (1). It proved to be useful and very efficient when considering subsonic or low-Mach number flows. In this case, the CFL restriction of Godunov-type schemes is driven by the acoustic waves and can be very restrictive. As we will see, theLagrange-Projection strategy allows fora very natural implicit-explicit scheme with a CFL restriction based on the (slow) transport waves and not on the (fast) acoustic waves, see the pionneering paper . Note that the low-Mach (or low-Froud in the present setting of shallowwaterequations) limit using the same techniques as in [4, 5, 6] will not be considered in the present paper but is the topic of current research. Here, we focus on the design of a high order well-balanced implicit explicit scheme in aLagrange-Projection framework.
In this work, we focus on the numerical approximation of theshallowwaterequations in two space dimensions. Our aim is to propose awell-balanced, all-regime and positive scheme. By well-balanced, it is meant that thescheme is able to preserve the so-called lake at rest smooth equilibrium solutions. By all-regime, we mean that thescheme 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, thescheme should be positive which means that thewater height stays positive for all time. Our approach is based on aLagrange- projection decomposition which allows to naturally decouple the acoustic and transport terms. Numerical experiments on unstructured meshes illustrate the good behaviour of thescheme.
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In this communication, we consider a numerical schemefortheshallow-water system. Thescheme under consideration has been proven to preserve the positivity of thewater height and to be fully well-balanced, i.e. to exactly preserve the smooth moving steady state solutions of theshallow-waterequations with the topography source term. The goal of this work is to prove a discrete entropy inequality satised by this scheme.
shallow-waterequations with topography
Christophe Berthon, Raphaël Loubère and Victor Michel-Dansac
Abstract We consider thewell-balanced numerical schemefortheshallow-waterequations with topography introduced in [ 8 ] and its second-order well-balanced extension, which requires two heuristic parameters. The goal of the present con- tribution is to derive a parameter-free second-order well-balancedscheme. To that end, we consider a convex combination between thewell-balancedscheme and a second-order scheme. We then prove that a relevant choice of the parameter of this convex combination ensures that the resulting scheme is both second-order accurate and well-balanced. Afterwards, we perform several numerical experiments, in order to illustrate both the second-order accuracy and thewell-balance property of this numerical scheme. Finally, we outline some perspectives in a short conclusion.
The steady states fortheshallow-waterequations with nonzero discharge are known to be more difficult to exactly capture than the lake at rest configuration. The critical role played by these specific solutions was illustrated in [ 32 ], were several benchmarks were exhibited. Next, in [ 30 ], a pioneer fully well-balancedscheme was designed to deal with these sensitive steady states, where the nu- merical technique is based on a suitable resolution of the Bernoulli equation. Next, several methods preserving the moving steady states were designed by involving high-order accurate techniques (see [ 16 , 42 , 49 , 51 ] for high-order and exactly well-balanced schemes, and [ 43 ] fora high-order accurate scheme on all steady state configurations).
We consider theshallow-waterequations 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-balancedscheme that approximates solutions of the system and preserves the associated steady states, including the moving ones. In addition, thescheme 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 thescheme while dealing with vanishing water heights. A second-order well-balanced MUSCL extension is designed, as well as an extension forthe two-dimensional case. Numerical experiments are performed in order to highlight the properties of thescheme. Keywords: shallow-waterequations, Manning friction, Godunov-type schemes, well-balanced schemes, moving steady states
These steady states are highly nonlinear, and exact preservation is a challenging task, see for instance [ 5 , 41 ] and references within. Note that, forthe 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 thewater height non-negativity, and the essential well- balancedness property. First steps have been undertaken in [ 11 ], where awell-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 awell-balanced high-order strategy, for both friction and topography source terms, that does not rely on solving nonlinear equations, and that is applicable to theshallowwaterequationsfor two-dimensional geometries.
The design of mixed implicit-explicit (IMEX) schemes based on aLagrange-Projection type ap- proach which are stable under a CFL restriction driven by the slow material waves and not the acoustic waves has been given a first interest in the pionneering work  and was further developed forthe computation of large friction or low-Mach regimes in , , , ,  for single or two-phase flow models. It is the purpose of this paper to adapt these IMEX strategies to theshallow-water equa- tions while preserving the first three properties above, namely the lake-at-rest well-balanced property, the positivity of thewater height, and the validity of a discrete form of the entropy inequality. An- other new large time step method fortheshallowwater flows in the low Froude number limit has been proposed in . The strategy is also mixed implicit-explicit considering the fast acoustic waves and the slow transport waves respectively, but does not rely on the natural Lagrange-Projection like decom- position proposed here. Note also that we focus here on subsonic or low Froude number flows, but we do not consider the low Froude number limit which is the purpose of a current work in progress. We also refer the reader to the recent contribution  which proves rigorously that the IMEX Lagrangeprojectionscheme is AP for one-dimensional low-Mach isentropic Euler and low-Froude shallowwaterequations.
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 ascheme able to capture the steady states is said Well-Balanced while ascheme able to restore the asymptotic regimes is said Asymptotic Preserv- ing. Thewell-balanced schemes were introduced in [ 6 , 18 ] in the framework of theshallow-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 ] fora 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-balancedscheme 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 ).
English version: We are interested in simulating blood flow in arter- ies with variable elasticity with a one dimensional model. We present awell-balanced finite volume scheme based on the recent developments in shallowwaterequations context. We thus get a mass conservative scheme which also preserves equilibria of Q = 0. This numerical method is tested on analytical tests.
The paper is organized as follows. Forthe sake of consistency of the present paper, in the next section we briefly recall the main algebraic properties satisfied by (1.1) and we emphasize that the steady states (1.3) can be understood as specific Riemann invariants of the model. In addition, we exhibit an unconsistency coming from thediscontinuous steady states and we thus give a precise definition of these equilibrium states of interest. In Section 3, we propose several remarks about the derivation of ap- proximate Riemann solvers and the associated Godunov-type schemes. In particular, we recall that the (exact) Godunov scheme obviously satisfies all the required prop- erties but fora non-analytical solvable Riemann problem. Therefore, we introduce our strategy by considering an approximate Riemann solver made of constant states including the topography source term in a sense to be specified. In fact, the relevant introduction of the source term inside the approximate Riemann solver imposes to consider an additional nonlinear relation. Firstly, in Section 4, we propose a suitable linearization of this non-standard relation to easily obtain a fully well-balanced and positivity preserving scheme. We note that this first naive approach illustrates the relevance of our approximate Riemann solver including the source term in its defini- tion. Indeed, at this level, we have straighforwardly designed a numerical scheme with better property than the generalized hydrostatic reconstruction. However, the stabil- ity of this first approach is not established. In order to derive an entropy preserving scheme, we suggest to consider the new nonlinear relation coming from the source term in its full generality. Then, Section 5 is devoted to the study of this relation and the characterization of the adopted approximate Riemann solver. Finally, we establish all the required properties; namely fully well-balanced property, positivity preserving and entropy preserving.
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 theshallow-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 theequations.
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 shallowwater type equations . Besides, also looking at the numerical approximation, dealing with vacuum needs fora special care, since we have to guarantee forthe non negativity of the solutions . 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.
January 30, 2012
In the last decades, more or less complex physically-based hydrologi- cal models, have been developed that solve theshallowwaterequations 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 schemefor mod- elling runoff. We present four different codes, each one based on either shallowwater 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 shallowwaterequations solved with the finite volume method.
The design of moving mesh methods on the sphere is a relatively recent subject of research. Optimal transport method have been discussed in [58, 59] using finite differences or finite-elements to solve the WeB:16,McR:18 Monge-Ampere type nonlinear PDEs involved. The results shown in terms of mesh quality and mesh refinement are very interesting. It is however unclear whether the error reduction compensates the CPU time overhead of mesh adaptation. Based on our previous experience, here we have opted fora simpler version of mesh movement on the sphere, trying to remain as much as possible close to the simplicity of ( 68). It is known that functional ( eq:mmpde-ch eq:mmpde-ch 69) can be generalized to model a mapping A : S eq:efun-ch eq:efun-ch A 2
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 thewater 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.
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-balancedscheme is a mathematical issue. In this work are presented new numerical results forthe spectrally well-balancedscheme in the context of highly irregular opacities.
DENYS DUTYKH ∗ AND DIDIER CLAMOND
Abstract. In this study, we propose an improved version of the nonlinear shallowwater (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 shallowwater ansatz and imposing suitable constraints. Thus, the derivation procedure does not explicitly involve any small parameter.
Christophe Berthon, Matthieu de Leffe, Victor Michel-Dansac
Abstract Ascheme defined by a hybridization between SPH method and finite volume method is considered. The aim of the present communication is to derive a suitable discretization of the source term to enforce the required well-balanced property. To address such an issue, we adopt a relevant reformulation of the flux function by involving the free surface instead of thewater height. Such an approach gives a natural discretization of the topography source term in order to preserve the lake at rest. Moreover, we prove that thescheme is in conservative form, which is, in general, a very difficult task since we do not impose restrictive assumptions on the SPH method. Several 1D numerical experiments are performed to exhibit the properties of thescheme.
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  or perfectly matched layers . The second equation in (1b) models the boundary of a perfectly conducting material. Finally, initial conditions are imposed in Ω