Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws

Bouchut, François

doi:10.1007/b93802

Cited by 506 publications

(907 citation statements)

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“…We give hereafter two general results which are satisfied for the particular case of the hydrostatic reconstruction introduced in [1] for the shallow water system with source terms. In fact, their proofs follow closely those of the corresponding results given in [3].…”

Section: Generalized Hydrostatic Reconstructionsupporting

confidence: 74%

“…Definition 2.1 generalizes the usual concept of well-balanced scheme for a system of balance laws (see [3]). It is a rather formal and geometrical definition, but it is well suited to the analysis of the well-balanced properties of path-conservative numerical schemes as these properties are strongly connected to the relationship between the paths and the set of curves Γ; see [20].…”

Section: Preliminariesmentioning

confidence: 99%

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

Castro¹,

Pardo²,

Parés³

et al. 2009

Math. Comp.

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Abstract. 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 wellbalanced 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.

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Section: Generalized Hydrostatic Reconstructionsupporting

confidence: 74%

Section: Preliminariesmentioning

confidence: 99%

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

Castro¹,

Pardo²,

Parés³

et al. 2009

Math. Comp.

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show abstract

“…This well-known difficulty has given rise to 'well-balanced schemes', which can accurately resolve balanced steady states, or small perturbations to such (e.g. [39,[42][43][44][45][46][47]). Prevalent equilibria of interest depend on the particular application.…”

Section: Numerical Methods and D-claw Softwarementioning

confidence: 99%

A depth-averaged debris-flow model that includes the effects of evolving dilatancy. II. Numerical predictions and experimental tests

2014

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We evaluate a new depth-averaged mathematical model that is designed to simulate all stages of debris-flow motion, from initiation to deposition. A companion paper shows how the model's five governing equations describe simultaneous evolution of flow thickness, solid volume fraction, basal pore-fluid pressure and two components of flow momentum. Each equation contains a source term that represents the influence of state-dependent granular dilatancy. Here, we recapitulate the equations and analyse their eigenstructure to show that they form a hyperbolic system with desirable stability properties.To solve the equations, we use a shock-capturing numerical scheme with adaptive mesh refinement, implemented in an open-source software package we call D-Claw. As tests of D-Claw, we compare model output with results from two sets of largescale debris-flow experiments. One set focuses on flow initiation from landslides triggered by rising pore-water pressures, and the other focuses on downstream flow dynamics, runout and deposition. D-Claw performs well in predicting evolution of flow speeds, thicknesses and basal pore-fluid pressures measured in each type of experiment. Computational results illustrate the critical role of dilatancy in linking coevolution of the solid volume fraction and porefluid pressure, which mediates basal Coulomb friction and thereby regulates debris-flow dynamics.

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“…The initial data for the swarmer density and for the surfactin concentration are given by n 0 = χ [1. 5,2] and, respectively, by the stationary solution of the surfactin equation with n replaced by n 0 . Other model parameters are given in the figure caption.…”

Section: Reduced 1d Model Without Supportersmentioning

confidence: 99%

Waves for a Hyperbolic Keller–segel Model and Branching Instabilities

Cerreti

Perthame

Schmeiser

et al. 2011

Math. Models Methods Appl. Sci.

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Recent experiments for swarming of the bacteria Bacillus subtilis on nutrient-rich media show that these cells are able to proliferate and spread out in colonies exhibiting complex patterns as dendritic ramifications. Is it possible to explain this process with a model that does not use local nutrient depletion? We present a new class of models which is compatible with the experimental observations and which predict branching instabilities and does not use nutrient limitation. These conclusions are based on numerical simulations. The most complex of these models is also the biologically most accurate but the essential effects can also be obtained in simplified versions which are amenable to analysis. An example of instability mechanism is the transition from a shock wave to a rarefaction wave in a reduced two by two hyperbolic system.

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Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws

Cited by 506 publications

References 0 publications

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

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

A depth-averaged debris-flow model that includes the effects of evolving dilatancy. II. Numerical predictions and experimental tests

Waves for a Hyperbolic Keller–segel Model and Branching Instabilities

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