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A well‐balanced scheme to capture non‐explicit steady states in the Euler equations with gravity
Author(s) -
Desveaux Vivien,
Zenk Markus,
Berthon Christophe,
Klingenberg Christian
Publication year - 2015
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.4177
Subject(s) - riemann solver , discretization , euler equations , mathematics , shallow water equations , finite volume method , multigrid method , nonlinear system , hydrostatic equilibrium , riemann problem , numerical analysis , relaxation (psychology) , solver , mathematical analysis , partial differential equation , backward euler method , riemann hypothesis , physics , mathematical optimization , mechanics , psychology , social psychology , quantum mechanics
Summary This paper describes a numerical discretization of the compressible Euler equations with a gravitational potential. A pertinent feature of the solutions to these inhomogeneous equations is the special case of stationary solutions with zero velocity, described by a nonlinear partial differential equation, whose solutions are called hydrostatic equilibria. We present a well‐balanced method, meaning that besides discretizing the complete equations, the method is also able to maintain all hydrostatic equilibria. The method is a finite volume method, whose Riemann solver is approximated by a so‐called relaxation Riemann solution that takes all hydrostatic equilibria into account. Relaxation ensures robustness, accuracy, and stability of our method, because it satisfies discrete entropy inequalities. We will present numerical examples, illustrating that our method works as promised. Copyright © 2015 John Wiley & Sons, Ltd.