A low dissipation method to cure the grid-aligned shock instability

The grid-aligned shock instability prevents an accurate computation of high Mach number flows using low-dissipation shock-capturing methods. In particular one manifestation, the so-called carbuncle phenomenon, has been investigated by various different groups over the past decades. Nevertheless, the mechanism of this instability is still not fully understood and commonly is suppressed by the introduction of additional numerical dissipation. However, present approaches may either significantly deteriorate the resolution of complex flow evolutions or involve additional procedures to limit stabilization measures to the shock region. Instead of increasing the numerical dissipation, in this paper, we present an alternative approach that relates the problem to the low Mach number in transverse direction of the shock front. We show that the inadequate scaling of the acoustic dissipation in the low Mach number limit is the prime reason for the instability. Our approach is to increase the “numerical” Mach number locally whenever the advection dissipation is small compared to the acoustic dissipation. A very simple modification of the eigenvalue calculation in the Roe approximation leads to a scheme with less numerical dissipation than the original Roe flux which prevents the grid-aligned shock instability. The simplicity of the modification allows for a detailed investigation of multidimensional effects. By showing that modifications in flow direction affect the shock stability in the transverse directions we confirm the multidimensional nature of the instability. The efficiency and robustness of the modified scheme is demonstrated for a wide range of test cases that are known to be particularly prone to the shock instability. Moreover, the modified flux also is successfully applied to multi-phase flows.