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The Method of Transport for solving the Euler-equations

by M. Fey

(Report number 1995-15)

Abstract
In many technical applications it is necessary to compute a numerical solution of complex flow problems in several space dimensions. Most available codes split the multi-dimensional problem into several one-dimension\-al ones. Those are aligned with the cell interfaces of the underlying grid. In some of the applications, e.g.high Mach number flow, this approach does not work very well, since the physical properties of the model equations are not represented correctly. In this paper a new idea to solve the multi-dimensional Euler equations numerically is presented. It is the aim of this paper to obtain a robust shock capturing method without the use of dimensional splitting and to get a better understanding of multi-dimensional phenomena. The starting point of this idea is the one-dimensional flux vector splitting and the homogeneity of the Euler equations. Using this concept it is shown that a different interpretation of the one-dimensional waves and the use of the characteristic surfaces lead to a decomposition of the state vector into three multi-dimensional waves. This idea includes the physical properties of the linearized Euler equations, i.e.it allows infinitely many propagation directions. Numerical results are shown at the end of this paper. It turns out that in special test cases, the multi-dimensional approach shown here and the dimensional splitting approach lead to structural differences even in a first order calculation.

Keywords: conservation laws, flux vector splitting, characteristic surfaces

@Techreport{F95_182,
  author = {M. Fey},
  title = {The Method of Transport for solving the Euler-equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {1995-15},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports1995/1995-15.pdf },
  year = {1995}
}

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