Systems of conservation laws

Lax, Peter D.; Wendroff, Burton

doi:10.1002/cpa.3160130205

Cited by 2,181 publications

(1,133 citation statements)

References 9 publications

(5 reference statements)

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“…[17,[30][31][32][33]). Most notably, the equations allow the development of non-unique discontinuous solutions requiring suitable numerical schemes that can converge stably to the correct solution.…”

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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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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“…The Lax-Wendroff Difference Scheme. In this section we shall be concerned with three versions of the Lax-Wendroff (L-W) difference scheme [6]. One may think of the L-W scheme as having an "artificial viscosity" built into it [7].…”

Section: Results Of Numerical Computationsmentioning

confidence: 99%

“…This was first noticed by Dr. S. Burstein in a calculation involving two space dimensions [9]. 6. The Navier-Stokes Equations.…”

Section: Ax2mentioning

confidence: 92%

On certain finite difference schemes for hyperbolic systems

Gary¹

1964

Math. Comp.

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1. Introduction. The purpose of this paper is to describe a few finite difference schemes for the numerical solution of hyperbolic systems of partial differential equations. Our main concern will be the stability conditions for these schemes although some of these schemes may be useful, especially for problems involving supersonic fluid flow. We will describe numerical experiments designed to check the derived stability conditions and also the accuracy of these schemes.In Section 2 we will describe the application of the Crank-Nicholson scheme to hyperbolic systems. This method is unconditionally stable, however it requires the inversion of a block tridiagonal matrix. We describe two modifications which require only the inversion of a scalar tridiagonal matrix. We carry out a stability analysis which shows that these schemes are unconditionally stable only if the matrix of the system is positive definite (supersonic flow in the case of hydrodynamics). Results of numerical computations using all these schemes are described in Section 3. All our results are for problems in one space dimension. These methods can be generalized to two dimensions but we have no analysis to indicate that the generalizations will work. In Section 5 we give the results of fluid dynamics computations using three versions of the Lax-Wendroff difference scheme. The objective here is to determine if it is necessary to write the equations in conservation form in order to obtain good results when the flow contains a shock. In Section 6 an application of the Lax-Wendroff scheme to the Navier-Stokes equations is described. An empirical stability condition is obtained which is a combination of the usual hyperbolic and parabolic conditions. In Section 7 an explicit difference scheme related to the CrankNicholson scheme is given. This is an iterative scheme with a rather peculiar sta-

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“…As examples we mention the conservation of mass, which is the basic framework for modern "shock-capturing" methods, the class of conservative schemes in the sense of Lax-Wendroff [134]; Energy conservative schemes which are advocated in long-term calculations of weather prediction using the global circulation model [3]; entropy stable schemes are sought as a "faithful" computations of physically relevant shock-discontinuities [198,201], and numerical methods which preserve the integrals of motions for (completely) integrable systems [217,107,108,7].…”

Section: 4mentioning

confidence: 99%

A review of numerical methods for nonlinear partial differential equations

Tadmor

2012

Bull. Amer. Math. Soc.

142

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Abstract. Numerical methods were first put into use as an effective tool for solving partial differential equations (PDEs) by John von Neumann in the mid1940s. In a 1949 letter von Neumann wrote "the entire computing machine is merely one component of a greater whole, namely, of the unity formed by the computing machine, the mathematical problems that go with it, and the type of planning which is called by both." The "greater whole" is viewed today as scientific computation: over the past sixty years, scientific computation has emerged as the most versatile tool to complement theory and experiments, and numerical methods for solving PDEs are at the heart of many of today's advanced scientific computations. Numerical solutions found their way from financial models on Wall Street to traffic models on Main Street. Here we provide a bird's eye view on the development of these numerical methods with a particular emphasis on nonlinear PDEs.

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Systems of conservation laws

Cited by 2,181 publications

References 9 publications

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

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

On certain finite difference schemes for hyperbolic systems

A review of numerical methods for nonlinear partial differential equations

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