a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations
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a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations St´ephane Clain1 · Rapha¨el Loub`ere2 · Gaspar J. Machado1
Received: 23 December 2016 / Accepted: 26 July 2017 © Springer Science+Business Media, LLC 2017
Abstract We propose a new family of high order accurate finite volume schemes devoted to solve one-dimensional steady-state hyperbolic systems. High-accuracy (up to the sixth-order presently) is achieved thanks to polynomial reconstructions while stability is provided with an a posteriori MOOD method which controls the cell polynomial degree for eliminating non-physical oscillations in the vicinity of discontinuities. Such a procedure demands the determination of a detector chain to discriminate between troubled and valid cells, a cascade of polynomial degrees to be successively tested when oscillations are detected, and a parachute scheme corresponding to the last, viscous, and robust scheme of the cascade. Experimented on linear, Burgers’, and Euler equations, we demonstrate that the schemes manage to retrieve smooth solutions with optimal order of accuracy but also irregular solutions without spurious oscillations.
Communicated by: Helge Holden The original version of this article was revised: During typesetting figures 8 and 21 got corrupted and showed erroneous graph lines. Both figures were updated. Gaspar J. Machado
[email protected] St´ephane Clain [email protected] Rapha¨el Loub`ere [email protected] 1
Centre of Mathematics, University of Minho, Campus of Azur´em, 4800-058 Guimar˜aes, Portugal
2
CNRS and Institut de Math´ematiques de Bordeaux (IMB), Universit´e de Bordeaux, Talence, France
S. Clain et al.
Keywords Finite volume · MOOD · Very high order · Hyperbolic equations · Steady-state Mathematics Subject Classification (2010) 65N08 · 65Z05 · 76M12
1 Introduction Numerical approximations of the steady-state Euler system date back to the early seventies with the NASA Ames group [42]. The development of an implicit time stepping algorithm [8] with high-order finite difference methods [9] led to the first simulations for two- and three-dimensional complex geometries calculated on Illiac IV Computer [43–45]. The steady-state was achieved as the limit stage of the nonstationary problem using a fictitious time step (time marching method) [50]. The main difficulties are, on one hand, to achieve an accurate approximation where the solution is smooth enough, and, on the other hand, to produce a robust solution without non-physical oscillations. Most of the high-order technologies developed for the non-stationary case were adapted to the steady-state case with the time marching algorithm using an explicit scheme [27] or an implicit formulation [33, 53] equipped with a Newton-Krylov iterative procedure [40, 48, 50] with extension to the full compressible or incompressible Navier-Stokes equations [55]. Taking the time step to infinity leads to a Newton-Krylov method for the steady-state problem which represents an alternativ
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