Carlos Parés

Numerical methods for nonconservative hyperbolic systems: a theoretical framework.

The goal of this paper is to provide a theoretical framework allowing one to extend some general concepts related to the numerical approximation of 1-d conservation laws to the more general case of first order quasi-linear hyperbolic systems. In particular this framework is intended to be useful for the design and analysis of well-balanced numerical schemes for solving balance laws or coupled systems of conservation laws. First, the concept of path-conservative numerical schemes is introduced, which is a generalization of the concept of conservative schemes for systems of conservation laws. Then, we introduce the general definition of approximate Riemann solvers and give the general expression of some well-known families of schemes based on these solvers: Godunov, Roe, and relaxation methods. Finally, the general form of a high order scheme based on a first order path-conservative scheme and a reconstruction operator is presented.The goal of this paper is to provide a theoretical framework allowing one to extend some general concepts related to the numerical approximation of 1-d conservation laws to the more general case of first order quasi-linear hyperbolic systems. In particular this framework is intended to be useful for the design and analysis of well-balanced numerical schemes for solving balance laws or coupled systems of conservation laws. First, the concept of path-conservative numerical schemes is introduced, which is a generalization of the concept of conservative schemes for systems of conservation laws. Then, we introduce the general definition of approximate Riemann solvers and give the general expression of some well-known families of schemes based on these solvers: Godunov, Roe, and relaxation methods. Finally, the general form of a high order scheme based on a first order path-conservative scheme and a reconstruction operator is presented.

High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems

This paper is concerned with the development of high order methods for the numerical approximation of one-dimensional nonconservative hyperbolic systems. In particular, we are interested in high order extensions of the generalized Roe methods introduced by I. Toumi in 1992, based on WENO reconstruction of states. We also investigate the well-balanced properties of the resulting schemes. Finally, we will focus on applications to shallow-water systems.

On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas

We present a finite volume scheme for solving shallow water equations with source term due to the bottom topography. The scheme has the following properties: it is high-order accurate in smooth wet regions, it correctly solves situations where dry areas are present, and it is well-balanced. The scheme is developed within a general nonconservative framework, and it is based on hyperbolic reconstructions of states. The treatment of wet/dry fronts is carried out by solving specific nonlinear Riemann problems at the corresponding intercells.

High Order Extensions of Roe Schemes for Two-Dimensional Nonconservative Hyperbolic Systems

This paper is concerned with the development of well-balanced high order Roe methods for two-dimensional nonconservative hyperbolic systems. In particular, we are interested in extending the methods introduced in (Castro et al., Math. Comput. 75:1103–1134, 2006) to the two-dimensional case. We also investigate the well-balance properties and the consistency of the resulting schemes. We focus in applications to one and two layer shallow water systems.

Well-balanced high-order numerical schemes for one-dimensional blood flow in vessels with varying mechanical properties

We construct well-balanced, high-order numerical schemes for one-dimensional blood flow in elastic vessels with varying mechanical properties. We adopt the ADER (Arbitrary high-order DERivatives) finite volume framework, which is based on three building blocks: a first-order monotone numerical flux, a non-linear spatial reconstruction operator and the solution of the Generalised (or high-order) Riemann Problem. Here, we first construct a well-balanced first-order numerical flux following the Generalised Hydrostatic Reconstruction technique. Then, a conventional non-linear spatial reconstruction operator and the local solver for the Generalised Riemann Problem are modified in order to preserve well-balanced properties. A carefully chosen suit of test problems is used to systematically assess the proposed schemes and to demonstrate that well-balanced properties are mandatory for obtaining correct numerical solutions for both steady and time-dependent problems.

On some fast well-balanced first order solvers for nonconservative systems

The goal of this article is to design robust and simple first order explicit solvers for one-dimensional nonconservative hyperbolic systems. These solvers are intended to be used as the basis for higher order methods for one or multidimensional problems. The starting point for the development of these solvers is the general definition of a Roe linearization introduced by Toumi in 1992 based on the use of a family of paths. Using this concept, Roe methods can be extended to nonconservative systems. These methods have good well-balanced and robustness properties, but they have also some drawbacks: in particular, their implementation requires the explicit knowledge of the eigenstructure of the intermediate matrices. Our goal here is to design numerical methods based on a Roe linearization which overcome this drawback. The idea is to split the Roe matrices into two parts which are used to calculate the contributions at the cells to the right and to the left, respectively. This strategy is used to generate two different one-parameter families of schemes which contain, as particular cases, some generalizations to nonconservative systems of the well-known Lax-Friedrichs, Lax-Wendroff, FORCE, and GFORCE schemes. Some numerical experiments are presented to compare the behaviors of the schemes introduced here with Roe methods.

NUMERICAL TREATMENT OF WET/DRY FRONTS IN SHALLOW FLOWS WITH A MODIFIED ROE SCHEME

This paper deals with the analysis of some numerical difficulties related to the appearance of wet/dry fronts that may occur during the simulation of free-surface waves in shallow fluids. The fluid is supposed to be governed by the Shallow Water equations and the discretization of the equations is performed, when wet/dry fronts do not appear, by means of the Q-scheme of Roe upwinding the source terms introduced in Ref. 40. This scheme is well-balanced in the sense that it solves exactly stationary solutions corresponding to water at rest. Wet/dry fronts cannot be correctly treated with this scheme: it can produce negative values of the thickness of the fluid layer and stationary solutions corresponding to water at rest including wet/dry transitions are not exactly solved. In Refs. 3–5 some variants of this numerical scheme have been proposed that partially solve these difficulties. Here we propose a new variant: at intercells where wet/dry transitions occur, a Nonlinear Riemann Problem is considered instead of a Linear one. The exact solutions of these nonlinear problems, which are easy to calculate, are used in order to define the numerical fluxes. We investigate the properties of the resulting scheme and present some comparisons with the numerical results obtained with some other modified numerical schemes proposed previously.

Well-Balanced High Order Extensions of Godunov's Method for Semilinear Balance Laws

This paper is concerned with the development of well-balanced high order numerical schemes for systems of balance laws with a linear flux function, whose coefficients may be variable. First, well-balanced first order numerical schemes are obtained based on the use of exact solvers of Riemann problems that include both the flux and the source terms. Godunovs methods so obtained are extended to higher order schemes by using a technique of reconstruction of states. The main contribution of this paper is to introduce a reconstruction technique that preserves the well-balanced property of Godunovs methods. Some numerical experiments are presented to verify in practice the properties of the developed numerical schemes.

High order exactly well-balanced numerical methods for shallow water systems

Abstract This paper is concerned with the development of high-order methods for the shallow water system that preserve every stationary solution of the system and not only the water-at-rest ones. The pursued strategy is based on the generalized hydrostatic reconstruction introduced in [10] , and on the technique proposed in [8] to obtain high-order well-balanced reconstruction operators. The more difficult step in the reconstruction procedure is the first one: given a cell value, a stationary solution of the system has to be found whose average at the cell equals the given value. The implementation of this step is discussed in detail, as well as, the similarities and differences with some previous approaches to obtain well-balanced methods for the shallow water system. Finally, the well-balanced properties of the developed numerical methods are checked in some test cases.

On the Convergence and Well-Balanced Property of Path-Conservative Numerical Schemes for Systems of Balance Laws

This paper deals with the numerical approximation of one-dimensional hyperbolic systems of balance laws. We consider these systems as a particular case of hyperbolic systems in nonconservative form, for which we use the theory introduced by Dal Maso, LeFloch and Murat (J. Math. Pures Appl. 74:483, 1995) in order to define the concept of weak solutions. This theory is based on the prescription of a family of paths in the phases space. We also consider path-conservative schemes, that were introduced in Parés (SIAM J. Numer. Anal. 44:300, 2006). The first goal is to prove a Lax-Wendroff type convergence theorem. In Castro et al. (J. Comput. Phys. 227:8107, 2008) it was shown that, for general nonconservative systems a rather strong convergence assumption is needed to prove such a result. Here, we prove that the same hypotheses used in the classical Lax-Wendroff theorem are enough to ensure the convergence in the particular case of systems of balance laws, as the numerical results shown in Castro et al. (J. Comput. Phys. 227:8107, 2008) seemed to suggest. Next, we study the relationship between the well-balanced properties of path-conservative schemes applied to systems of balance laws and the family of paths.