The numerical solution of hyperbolic systems of partial differential equations in three independent variables

Butler, Dagny

doi:10.1098/rspa.1960.0065

Cited by 107 publications

(18 citation statements)

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“…Section 2 will be devoted to finite difference and evolution Galerkin methods for the wave equation system which are based on the straightforward use of the bicharacteristic cone. We will describe the approach of Butler [3] (and its follow up by Prasad et al [25], [26]) who first used bicharacteristics in order to derive numerical schemes. We will prove a useful lemma which allows us to derive new approximate evolution operators for the wave equation system.…”

Section: Introductionmentioning

confidence: 99%

Evolution Galerkin methods for hyperbolic systems in two space dimensions

2000

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Abstract. The subject of the paper is the analysis of three new evolution Galerkin schemes for a system of hyperbolic equations, and particularly for the wave equation system. The aim is to construct methods which take into account all of the infinitely many directions of propagation of bicharacteristics. The main idea of the evolution Galerkin methods is the following: the initial function is evolved using the characteristic cone and then projected onto a finite element space. A numerical comparison is given of the new methods with already existing methods, both those based on the use of bicharacteristics as well as commonly used finite difference and finite volume methods. We discuss the stability properties of the schemes and derive error estimates.

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Section: Introductionmentioning

confidence: 99%

Evolution Galerkin methods for hyperbolic systems in two space dimensions

2000

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“…Following these authors, we will call their integral representation the evolution Galerkin, or EG operator. This operator is based on classical characteristic theory, see [3,5,28].…”

Section: Comparison With the Eg Evolution Operatormentioning

confidence: 99%

“…• the evolution Galerkin (EG) approach of Butler [3], Morton et al [19] (exploiting the transport collapse operator of Brenier [2]), Ostkamp [24,25], Lukáčová, Morton, Warnecke [20] as well as (based on this) the finite volume evolution Galerkin (FVEG) approach of Lukáčová, Morton, Saibertová and Warnecke [21,22]; and • the kinetic approach of Deshpande [6] and Perthame [26,27].…”

Section: Introductionmentioning

confidence: 99%

On the connection between some Riemann-solver free approaches to the approximation of multi-dimensional systems of hyperbolic conservation laws

Kröger¹,

Noelle²,

Zimmermann

2004

ESAIM: M2AN

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Abstract.In this paper, we present some interesting connections between a number of Riemannsolver free approaches to the numerical solution of multi-dimensional systems of conservation laws. As a main part, we present a new and elementary derivation of Fey's Method of Transport (MoT) (respectively the second author's ICE version of the scheme) and the state decompositions which form the basis of it. The only tools that we use are quadrature rules applied to the moment integral used in the gas kinetic derivation of the Euler equations from the Boltzmann equation, to the integration in time along characteristics and to space integrals occurring in the finite volume formulation. Thus, we establish a connection between the MoT approach and the kinetic approach. Furthermore, Ostkamp's equivalence result between her evolution Galerkin scheme and the method of transport is lifted up from the level of discretizations to the level of exact evolution operators, introducing a new connection between the MoT and the evolution Galerkin approach. At the same time, we clarify some important differences between these two approaches.Mathematics Subject Classification. 35C15, 35L65, 65D32, 65M25, 76M12, 76N15, 76P05.

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“…They provide utmost correspondence between the dependence regions of the starting differential equations and approximating difference equations, what notably increases the accuracy of results for smooth and discontinuous solutions; they also provide correct identification of boundaries and contacts. In 1960, an explicit scheme of second order was suggested for a system of partial differential equations of second order in three variables [1]. The scheme employed characteristics and it was used for studying plane waves [2].…”

Section: Introductionmentioning

confidence: 99%

In uence of boundary conditions on 2D wave propagation in a rectangle

Ashirbayev¹,

Ashirbayeva²

2013

JMA

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Work is devoted to generalization of a differential method of spatial characteristics to case of the flat task about distribution of waves in rectangular area of the final sizes with gaps in boundary conditions. On the basis of the developed numerical technique are received the settlement certainly -differential ratios of dynamic tasks in special points of front border of rectangular area, where boundary conditions on coordinate aren't continuous. They suffer a rupture of the first sort in points in which action P -figurative dynamic loading begins. Results of research are brought to the numerical decision.

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The numerical solution of hyperbolic systems of partial differential equations in three independent variables

Cited by 107 publications

References 0 publications

Evolution Galerkin methods for hyperbolic systems in two space dimensions

Evolution Galerkin methods for hyperbolic systems in two space dimensions

On the connection between some Riemann-solver free approaches to the approximation of multi-dimensional systems of hyperbolic conservation laws

In uence of boundary conditions on 2D wave propagation in a rectangle

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