Stability analysis of the carbuncle phenomenon and the sonic point glitch

Abstract

Upwinding allows for accurate, non-oscillatory capturing of shocks waves; however, many Riemann solvers (both exact and approximate) suffer from some sort of numerical instability. One of the most mysterious and least understood of these is the carbuncle phenomenon. In the present study, we analyse the closely allied “simplified carbuncle” problem, also known as the 2D shock stability problem or the 1.5D carbuncle problem. Motivated by the existence of some recently derived schemes that do not exhibit the instability, we perform a thorough stability analysis and extend previous studies by analysing the pseudo-spectra and hence the effects of non-normality in causing this instability. Our results establish that, contrary to previous indications in the literature, a non-linear mechanism is responsible for the instability. In order to understand the nature of this non-linear mechanism better, we perform a non-linear analysis of the sonic glitch, which shares some common features with the carbuncle. We provide two previously unknown results. Firstly, we show that even the “entropy-satisfying” Godunov scheme violates the entropy condition in the sonic glitch. Secondly, we provide a more accurate definition for the entropy condition for scalar conservation laws that supports the previous claim. We conjecture that a similar non-linear anti-dissipative mechanism might be responsible in triggering the carbuncle. This work is expected to lead to a better understanding of possible unphysical behaviour in Riemann solvers and thus help in the design of better solvers for high-Reynolds-number flows.