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Multidimensional schemes for nonlinear systems of hyperbolic conservation laws

by M. Fey and R. Jeltsch and A.-T. Morel

(Report number 1995-11)

Abstract
Most commonly used schemes for unsteady multidimensional systems of hyperbolic conservation laws use dimensional splitting. In each coordinate direction a scheme for a one dimensional system is used. Such an approach does not take in account the infinitely many propagation directions which are present in a system in several space dimensions. In 1992 M. Fey introduced what he called the Method of Transport, MoT, for the Euler equations of gas dynamics. It is a finite volume method which uses the transport along characteristics. It does not compute fluxes across cellsides but from one cell to another. These type of schemes can be developed by first rewriting the Euler equation as a sum with integrals of infinitely many transport equations. One of these terms is related to the transport by the velocity while the integrals reflect the acoustic waves. In the numerical scheme the integrals are replaced by finite sums. The method can be modified such as to become a second order scheme. The technique can be applied to the magneto-hydrodynamic equations and the shallow water equation. Numerical examples for the shallow water equation are given.

Keywords: nonlinear hyperbolic conservation laws, multi-dimensional schemes, method of transport, second order, Euler equations of gas dynamics, shallow water equation, magneto-hydrodynamic equations

@Techreport{FJM95_178,
  author = {M. Fey and R. Jeltsch and A.-T. Morel},
  title = {Multidimensional schemes for nonlinear systems of hyperbolic conservation laws},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {1995-11},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports1995/1995-11.pdf },
  year = {1995}
}

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