In this paper, we propose a new class of second-order one-step time-splitting schemes for scalar and systems of conservation laws. The strategy is based on a relaxation approximation which consists in introducing a relaxation system with linearly degenerate characteristic fields to approximate the original system and deal with its nonlinearities. The numerical schemes are based on the use of arbitrarily high-order approximations of the underlying linear advection equations. Numerical evidence is proposed to illustrate the behaviour of our schemes.

In this work we develop a new framework to deal numerically with discontinuous solutions in nonconservative hyperbolic systems. First an extension of the MOOD methodology to nonconservative systems based on Taylor expansions is presented. This extension combined with an in-cell discontinuous reconstruction operator are the key points to develop a new family of high-order methods that are able to capture exactly isolated shocks. Several test cases are proposed to validate these methods for the Modified Shallow Water equations and the Two-Layer Shallow Water system.

This article focuses on the design of semi-implicit schemes that are fully well-balanced for the one-dimensional shallow water equations, that is, schemes that preserve all smooth steady states of the system and not just water-at-rest equilibria. The proposed methods outperform standard explicit schemes in the low-Froude regime, where the celerity is much larger than the fluid velocity, eliminating the need for a large number of iterations on large time intervals. In this work, splitting and relaxation techniques are combined in order to obtain fully well-balanced semi-implicit first and second order schemes. In contrast to recent Lagrangian-based approaches, this one allows the preservation of all the steady states while avoiding the complexities associated with Lagrangian formalism.

We present a numerical approximation of the solutions of the Euler equations with a gravitational source term. On the basis of a Suliciu type relaxation model with two relaxation speeds, we construct an approximate Riemann solver, which is used in a first order Godunov-type finite volume scheme. This scheme can preserve both stationary solutions and the low Mach limit to the corresponding incompressible equations. In addition, we prove that our scheme preserves the positivity of density and internal energy, that it is entropy satisfying and also guarantees not to give rise to numerical checkerboard modes in the incompressible limit. Later we give an extension to second order that preserves positivity, asymptotic-preserving and well-balancing properties. Finally, the theoretical properties are investigated in numerical experiments.

This works concerns the study of well-balanced Lagrange-projection schemes applied to the two-layer shallow water system. In particular, a formulation of the mathematical model in Lagrangian coordinates is proposed. The HLL method is then applied to a simplified version of the resulting Lagrangian system. Furthermore, based on the acoustic-transport splitting interpretation, another approximate Riemann solver for the acoustic-Lagrangian step is described. Both an explicit and an implicit-explicit method are proposed, where the latter can allow very fast simulations in sub-critical regimes. Finally, we show numerical simulations in which the outputs are compared with the IFCP method's results. (c) 2022 Elsevier Inc. All rights reserved.

In the present work we aim to simulate shallow water flows over movable bottom with suspended and bedload transport. In order to numerically approximate such a system, we proceed step by step. We start by considering shallow water equations with non-constant density of the mixture water-sediment. Then, the Exner equation is included to take into account bedload sediment transport. Finally, source terms for friction, erosion and deposition processes are considered. Indeed, observe that the sediment particle could go in suspension into the water or being deposited on the bottom. For the numerical scheme, we rely on well-balanced Lagrange-projection methods. In particular, since sediment transport is generally a slow process, we aim to develop semi-implicit schemes in order to obtain fast simulations. The Lagrange-projection splitting is well-suited for such a purpose as it entails a decomposition of the (fast) acoustic waves and the (slow) material waves of the model. Hence, in subsonic regimes, an implicit approximation of the acoustic equations allows us to neglect the corresponding CFL condition and to obtain fast numerical schemes with large time step.

The present work is devoted to the numerical approximation of the shallow water Exner system in both one and two dimensions, where the Exner equation expresses the evolution in time of the bed sediment. Both the Grass and the Meyer-Peter & Mu & BULL;ller formulas are taken into account to model the solid transport discharge contributions. The numerical scheme is based on the Lagrange-projection formalism which consists in splitting the mathematical model into the acoustic and transport systems. This work is considered as a first step to understand how to include the Exner equation in this framework and, for this reason, the Exner equation is taken into account only at the transport level; both a decoupled and weakly coupled formulations are proposed. New strategies to include the Exner equation at the acoustic level or in both steps will be treated in the next work. The methods are designed in such a way to satisfy the well-balanced property as well. Details to reach the second-order of accuracy are given; numerical results are shown to validate the numerical schemes.

This work is devoted to the numerical approximation of the shallow water Exner system. We investigate three different numerical strategies to discretize the Exner equation which expresses the evolution in time of the bed sediment. The numerical schemes are all based on the Lagrange-projection formalism which consists in splitting the mathematical model into the acoustic and transport system. In particular, the Exner equation is taken into account either in the acoustic and transport steps, or only at the acoustic or transport level. The methods and their second-order extensions are designed in such a way to satisfy the well-balanced property, namely the "lake at rest" and the "constant bed slope" steady states. Numerical evidences are given to validate the numerical schemes.

We are interested in the numerical approximation of discontinuous solutions in non conservative hyperbolic systems. An extension to second-order of a new strategy based on in-cell discontinuous reconstructions to deal with this challenging topic is presented. This extension is based on the combination of the first-order in-cell reconstruction and the MUSCL-Hancock reconstruction. The first-order strategy allowed in particular to capture exactly the isolated shocks and this new second-order extension keep this property. We also set the basis of an extension to high-order methods following this in-cell methodology and a way for capturing exactly more than one shock. Several numerical tests are proposed to validate the methods for the Coupled-Burgers system, Gas dynamics equations in Lagrangian coordinates and the modified shallow water system.

The importance of the study of the blood flow equations is widely recognized as it is a tool to understand the circulatory system. Arteries and veins result to have both elastic and viscous behaviour. Models for the first case are much more studied as they result to be simpler and still satisfying if compared to experimental data. In this paper, we consider a model which encompasses both the elastic and viscoelastic response in arterial walls, respectively leading to a conservative and a non-conservative system. We present a second order scheme based on the first-order Price-T scheme and the MUSCL-Hancock strategy. This approach automatically adapts to the above conservative and non-conservative cases. Then, we perform a Sensitivity Analysis (SA) based on the Continuous Sensitivity Equation Method (CSEM), whose aim is the study of how changes in the inputs of a model can affect its outputs. In particular, the sensitivity is defined as the derivative (with respect to an uncertain parameter a) of the solution of the system taken into consideration. Since the CSEM cannot be directly applied to discontinuous solutions, we add a source term to compensate the spikes associated to the Dirac delta functions that can arise in the sensitivity variables. One of the main applications of SA is uncertainty quantification, which is investigated for a Riemann problem as well as for a network of 37 arteries. Details on junctions for coupling two or more vessels are also given. (c) 2022 Elsevier Inc. All rights reserved.

We focus on the development of well-balanced Lagrange-projection schemes applied to the one-dimensional blood flow system of balance laws. Here we neglect the friction forces and the source term is due to the presence of varying parameters as the cross-sectional area at the equilibrium and the arterial stiffness. By well-balanced we mean that the method preserves the “man at eternal rest” solution. For this purpose we present two different strategies: the former requires a consistent definition of the source term based on an approximate Riemann solver, while the second one exploits the well-established hydrostatic reconstruction. Subsequently we explain how to reach the second-order of accuracy for both procedures. Numerical simulations are carried out in order to show the right order of accuracy and the good behaviour of the schemes.

The well-known Lighthill-Whitham-Richards (LWR) kinematic model of traffic flow models the evolution of the local density of cars by a nonlinear scalar conservation law. The transition between free and congested flow regimes can be described by a flux or velocity function that has a discontinuity at a determined density. A numerical scheme to handle the resulting LWR model with discontinuous velocity was proposed in [J.D. Towers, A splitting algorithm for LWR traffic models with flux discontinuities in the unknown, J. Comput. Phys., 421 (2020), article 109722]. A similar scheme is constructed by decomposing the discontinuous velocity function into a Lipschitz continuous function plus a Heaviside function and designing a corresponding splitting scheme. The part of the scheme related to the discontinuous flux is handled by a semi-implicit step that does, however, not involve the solution of systems of linear or nonlinear equations. It is proved that the whole scheme converges to a weak solution in the scalar case. The scheme can in a straightforward manner be extended to the multiclass LWR (MCLWR) model, which is defined by a hyperbolic system of N conservation laws for N driver classes that are distinguished by their preferential velocities. It is shown that the multiclass scheme satisfies an invariant region principle, that is, all densities are nonnegative and their sum does not exceed a maximum value. In the scalar and multiclass cases no flux regularization or Riemann solver is involved, and the CFL condition is not more restrictive than for an explicit scheme for the continuous part of the flux. Numerical tests for the scalar and multiclass cases are presented.

In this paper, we consider the numerical approximation of hyperbolic systems of conservation laws with stiff source terms and parabolic degeneracy in the asymptotic limit. We are more precisely interested in the design of high-order asymptotic-preserving schemes on unstructured meshes. Our approach is based on a very simple modification of the numerical flux associated with the usual HLL scheme and boils down to a sharp control of the underlying numerical diffusion. The strategy allows to capture the correct asymptotic parabolic behaviour and to preserve the high-order accuracy also in the asymptotic limit. Numerical experiments are proposed to illustrate these properties.

The continuous sensitivity equation method allows to quantify how changes in the input of a partial differential equation (PDE) model affect the outputs, by solving additional PDEs obtained by differentiating the model. However, this method cannot be used directly in the framework of hyperbolic PDE systems with discontinuous solution, because it yields Dirac delta functions in the sensitivity solution at the location of state discontinuities. This difficulty is well known from theoretical viewpoint, but only a few works can be found in the literature regarding the possible numerical treatment. Therefore, we investigate in this study how classical numerical schemes for compressible Euler equations can be modified to account for shocks when computing the sensitivity solution. In particular, we propose the introduction of a source term, that allows to remove the spikes associated to the Dirac delta functions in the numerical solution. Numerical studies exhibit a strong impact of the numerical diffusion on the accuracy of this strategy. Therefore, we propose an anti‐diffusive numerical scheme coupled with the approximate Riemann solver of Roe for the state problem. For the sensitivity problem, two different numerical schemes are implemented and compared: one which takes into account the contact wave and another that neglects it. The effects of the numerical diffusion on the convergence of the schemes with respect to the grid are discussed. Finally, an application to uncertainty propagation is investigated and the different numerical schemes are compared.

In this work we propose a novel strategy to define high-order fully well-balanced LagrangeProjection finite volume solvers for balance laws. In particular, we focus on the 1D shallow water system as it is a reference system of balance laws with non-trivial stationary solutions. Nevertheless, the strategy proposed here could be extended to other interesting balance laws. By fully well-balanced, it is meant that the scheme is able to preserve stationary smooth solutions. Following [6], we exploit the idea of using a high-order well-balanced reconstruction operator for the Lagrangian step. Nevertheless, this is not enough to achieve well-balanced high-order during the projection step. We propose here a new projection step that overcomes this difficulty and that reduces to the standard one in case of conservation laws. Finally, some numerical experiments illustrate the good behaviour of the scheme.

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.

In the first part of this work, we introduce a new relaxation system in order to approximate the solutions to the barotropic Euler equations. We show that the solutions to this two-speed relaxation model can be understood as viscous approximations of the solutions to the barotropic Euler equations under appropriate sub-characteristic conditions. Our relaxation system is a generalization of the well-known Suliciu relaxation system, and it is entropy satisfying. A Godunov-type finite volume scheme based on the exact resolution of the Riemann problem associated with the relaxation system is deduced, as well as its stability properties. In the second part of this work, we show how the new relaxation approach can be successfully applied to the numerical approximation of low Mach number flows. We prove that the underlying scheme satisfies the well-known asymptotic-preserving property in the sense that it is uniformly (first-order) accurate with respect to the Mach number, and at the same time it satisfies a fully discrete entropy inequality. This discrete entropy inequality allows us to prove strong stability properties in the low Mach regime. At last, numerical experiments are given to illustrate the behaviour of our scheme.

In this work, we address the numerical approximation of linear systems with possibly stiff source terms which induce an asymptotic diffusion limit. More precisely, we are interested in the design of high‐order asymptotic‐preserving schemes. Our approach is based on a very simple modification of the numerical flux associated with the usual HLL scheme. This alteration can be understood as a numerical diffusion reduction technique and allows to capture the correct asymptotic behavior in the diffusion limit and to consider uniformly high‐order extensions. We more specifically consider the case of the Goldstein–Taylor model but the overall approach is shown to be easily adapted to more general systems.

This paper focuses on the numerical approximation of the solutions of nonlocal conservation laws in one space dimension. These equations are motivated by two distinct applications, namely, a traffic flow model in which the mean velocity depends on a weighted mean of the downstream traffic density, and a sedimentation model where either the solid phase velocity or the solid-fluid relative velocity depends on the concentration in a neighborhood. In both models, the velocity is a function of a convolution product between the unknown and a kernel function with compact support. It turns out that the solutions of such equations may exhibit oscillations that are very difficult to approximate using classical first-order numerical schemes. We propose to design discontinuous Galerkin (DG) schemes and finite volume WENO (FV-WENO) schemes to obtain high-order approximations. As we will see, the DG schemes give the best numerical results but their CFL condition is very restrictive. On the contrary, FV-WENO schemes can be used with larger time steps. We will see that the evaluation of the convolution terms necessitates the use of quadratic polynomials reconstructions in each cell in order to obtain high-order accuracy with the FV-WENO approach. Simulations using DG and FV-WENO schemes are presented for both applications.

In this work, an asymptotic-preserving scheme is proposed for the electronic M-1 model in the diffusion limit. A very simple modification of the HLL numerical viscosity is considered in order to capture the correct asymptotic limit in the diffusion limit. This alteration also ensures the realizability of the numerical solution under a suitable CFL condition. Interestingly, it is proved that the new scheme can also be understood as a Godunov-type scheme based on a suitable approximate Riemann solver. Various numerical test cases are performed and the results are compared with a standard HLL scheme and an explicit discretization of the limit diffusion equation.