A New Robust Carbuncle-Free Roe Scheme for Strong Shock

The paper devises a new robust carbuncle-free Roe Riemann solver for strong shock, different from hybrid method, entropy fix, Liou’s conjecture (J Comput Phys 160:623–648, 2000) and artificial viscosity (Rodionov in J Comput Phys 345:308–329, 2017). Roe scheme encounters carbuncle phenomenon, violates entropy condition and lacks positivity property. The remedy integrates shear viscosity into momentum flux to damp undesirable perturbation and prevents unstable vorticity mode from triggering shock instability. Compared to HLL scheme, shear viscosity is properly established through dimensional analysis and analogy method. The non-linear wave speeds are slightly modified by incorporating the neighboring cell information for positive conservation. The pressure-based sensing function is applied to preserve shear layer while keeping shock robustness. The resulting scheme is very easily implemented by converting the existing Roe code. The matrix stability analysis confirms that this approach is shock-stable and more robust than entropy fix. A series of numerical results demonstrate its potential features: positivity-preserving property, entropy-satisfying property, accurate boundary-layer resolution and high robustness against shock instability. In addition, it signifies that the reason to cause shock instability for Roe scheme may be not the pressure difference term in mass flux, but the inadequate shear viscosity.