On the pathological behavior of upwind schemes

Abstract

Despite constant progress in the development of upwind schemes, some failings still remain. Quirk recently reported that approximate Riemann solvers, which share the exact capture of contact discontinuities, generally suffer from such failings. One of them is the odd-even decoupling that occurs along planar shocks aligned with the mesh. Quirk proposed to test this shortcoming with the propagation of a planar shock in a duct. First, we give a few results on some failings. Then, following Quirk's analysis of Roe's scheme, general criteria are derived to predict the odd-even decoupling. This analysis is applied to Roe's scheme, EFM Pullin's scheme, EIM Macrossan's scheme and AUSM Liou's scheme. Strict stability is shown to be desirable to avoid must of these flaws.

A Time-accurate Upwind Unstructured Finite volume Method for Compressible Flow with Cure of Pathological Behaviors

“…These pathological behaviors occur mostly when a grid line is aligned with a strong shock or expansion. It is generally agreed that insufficient dissipation and the consequent local numerical instability lead to such failures [12,13,15]. In the literature, for the Godunov type upwind schemes, a general cure is switching to different Riemann solvers to gain some more dissipation.…”

“…For example, an expansion shock can be cured by switching from a Roe approximate Riemann solver to an HLLE Riemann solver [12]. A comparison of the performances of various Riemann solvers or flux-splitting is given in [13]. One needs to know in advance which Riemann solver is appropriate for the particular flow problem being considered.…”

“…In this work a basic upwind scheme is chosen similar to the one presented, for solving the Euler equations, in [1]. As the upwind methods are prone to exhibit "pathological behaviors" [12][13][14][15][16] such as the carbuncle phenomenon and the expansion shock, the method presented here will provide a cure for these undesired phenomena.…”

Towards An 'All Speed' Unstructured Upwind Scheme

“…Numerical dissipation is required to stabilize the scheme. [11][12][13][14]. For extremely low Mach number flows, an upwind scheme often needs preconditioning treatment for a successful computation.…”

Near-Field Noise Computation for a Subsonic Coannular Jet

“…29 The method (ETAU) is nominally second-order accurate in space and time and is capable of yielding high resolution in the presence of sharp gradients. A built-in dissipation model helps to overcome potential pathological behaviors 30,31 of upwind schemes and renders the ETAU scheme robust and viable for practical computations.…”