A hybrid scheme to compute contact discontinuities in one-dimensional Euler systems

Thierry Gallouët; Jean-Marc Hérard; Nicolas Seguin

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 36, Issue: 6, page 1133-1159
  • ISSN: 0764-583X

Abstract

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The present paper is devoted to the computation of single phase or two phase flows using the single-fluid approach. Governing equations rely on Euler equations which may be supplemented by conservation laws for mass species. Emphasis is given on numerical modelling with help of Godunov scheme or an approximate form of Godunov scheme called VFRoe-ncv based on velocity and pressure variables. Three distinct classes of closure laws to express the internal energy in terms of pressure, density and additional variables are exhibited. It is shown first that a standard conservative formulation of above mentioned schemes enables to predict “perfectly” unsteady contact discontinuities on coarse meshes, when the equation of state (EOS) belongs to the first class. On the basis of previous work issuing from literature, an almost conservative though modified version of the scheme is proposed to deal with EOS in the second or third class. Numerical evidence shows that the accuracy of approximations of discontinuous solutions of standard Riemann problems is strengthened on coarse meshes, but that convergence towards the right shock solution may be lost in some cases involving complex EOS in the third class. Hence, a blend scheme is eventually proposed to benefit from both properties (“perfect” representation of contact discontinuities on coarse meshes, and correct convergence on finer meshes). Computational results based on an approximate Godunov scheme are provided and discussed.

How to cite

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Gallouët, Thierry, Hérard, Jean-Marc, and Seguin, Nicolas. "A hybrid scheme to compute contact discontinuities in one-dimensional Euler systems." ESAIM: Mathematical Modelling and Numerical Analysis 36.6 (2010): 1133-1159. <http://eudml.org/doc/194143>.

@article{Gallouët2010,
abstract = { The present paper is devoted to the computation of single phase or two phase flows using the single-fluid approach. Governing equations rely on Euler equations which may be supplemented by conservation laws for mass species. Emphasis is given on numerical modelling with help of Godunov scheme or an approximate form of Godunov scheme called VFRoe-ncv based on velocity and pressure variables. Three distinct classes of closure laws to express the internal energy in terms of pressure, density and additional variables are exhibited. It is shown first that a standard conservative formulation of above mentioned schemes enables to predict “perfectly” unsteady contact discontinuities on coarse meshes, when the equation of state (EOS) belongs to the first class. On the basis of previous work issuing from literature, an almost conservative though modified version of the scheme is proposed to deal with EOS in the second or third class. Numerical evidence shows that the accuracy of approximations of discontinuous solutions of standard Riemann problems is strengthened on coarse meshes, but that convergence towards the right shock solution may be lost in some cases involving complex EOS in the third class. Hence, a blend scheme is eventually proposed to benefit from both properties (“perfect” representation of contact discontinuities on coarse meshes, and correct convergence on finer meshes). Computational results based on an approximate Godunov scheme are provided and discussed. },
author = {Gallouët, Thierry, Hérard, Jean-Marc, Seguin, Nicolas},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Godunov scheme; Euler system; contact discontinuities; thermodynamics; conservative schemes.; thermodynamics; conservative schemes},
language = {eng},
month = {3},
number = {6},
pages = {1133-1159},
publisher = {EDP Sciences},
title = {A hybrid scheme to compute contact discontinuities in one-dimensional Euler systems},
url = {http://eudml.org/doc/194143},
volume = {36},
year = {2010},
}

TY - JOUR
AU - Gallouët, Thierry
AU - Hérard, Jean-Marc
AU - Seguin, Nicolas
TI - A hybrid scheme to compute contact discontinuities in one-dimensional Euler systems
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 36
IS - 6
SP - 1133
EP - 1159
AB - The present paper is devoted to the computation of single phase or two phase flows using the single-fluid approach. Governing equations rely on Euler equations which may be supplemented by conservation laws for mass species. Emphasis is given on numerical modelling with help of Godunov scheme or an approximate form of Godunov scheme called VFRoe-ncv based on velocity and pressure variables. Three distinct classes of closure laws to express the internal energy in terms of pressure, density and additional variables are exhibited. It is shown first that a standard conservative formulation of above mentioned schemes enables to predict “perfectly” unsteady contact discontinuities on coarse meshes, when the equation of state (EOS) belongs to the first class. On the basis of previous work issuing from literature, an almost conservative though modified version of the scheme is proposed to deal with EOS in the second or third class. Numerical evidence shows that the accuracy of approximations of discontinuous solutions of standard Riemann problems is strengthened on coarse meshes, but that convergence towards the right shock solution may be lost in some cases involving complex EOS in the third class. Hence, a blend scheme is eventually proposed to benefit from both properties (“perfect” representation of contact discontinuities on coarse meshes, and correct convergence on finer meshes). Computational results based on an approximate Godunov scheme are provided and discussed.
LA - eng
KW - Godunov scheme; Euler system; contact discontinuities; thermodynamics; conservative schemes.; thermodynamics; conservative schemes
UR - http://eudml.org/doc/194143
ER -

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