a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations

Stéphane Clain (1) , Raphaël Loubère (2, 3) , Gaspar J Machado (1)

1 CMAT - Centro de Matemática [Minho]
2 IMB - Institut de Mathématiques de Bordeaux
3 CNRS - Centre National de la Recherche Scientifique

We propose a new family of finite volume high-accurate numerical 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 control the cell polynomial degree for eliminating non-physical oscillations in the vicinity of discontinuities. Such a procedure demands the determination of a chain detector 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.

Mots clés: steady-state; hyperbolic equations; finite volume; high-order; MOOD

Domaines: Analyse numérique [math.NA]

Stéphane Clain, Raphaël Loubère, Gaspar J Machado. a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations. 2016. ⟨hal-01418899⟩