Artificial viscosity in Godunov-type schemes to cure the carbuncle phenomenon

This work presents a new approach for curing the carbuncle instability. The idea underlying the approach is to introduce some dissipation in the form of right-hand sides of the Navier-Stokes equations into the basic method of solving Euler equations; in so doing, we replace the molecular viscosity coefficient by the artificial viscosity coefficient and calculate heat conductivity assuming that the Prandtl number is constant. For the artificial viscosity coefficient we have chosen a formula that is consistent with the von Neumann and Richtmyer artificial viscosity, but has its specific features (extension to multidimensional simulations, introduction of a threshold compression intensity that restricts additional dissipation to the shock layer only). The coefficients and the expression for the characteristic mesh size in this formula are chosen from a large number of Quirk-type problem computations. The new cure for the carbuncle flaw has been tested on first-order schemes (Godunov, Roe, HLLC and AUSM+ schemes) as applied to one- and two-dimensional simulations on smooth structured grids. Its efficiency has been demonstrated on several well-known test problems.