Realization of contact resolving approximate Riemann solvers for strong shock and expansion flows

The Harten-Lax-van Leer contact (HLLC) and Roe schemes are good approximate Riemann solvers that have the ability to resolve shock, contact, and rarefaction waves. However, they can produce spurious solutions, called shock instabilities, in the vicinity of strong shock. In strong expansion flows, the Roe scheme can admit nonphysical solutions such as expansion shock, and it sometimes fails. We carefully examined both schemes and propose simple methods to prevent such problems. High-order accuracy is achieved using the weighted average flux (WAF) and MUSCL-Hancock schemes. Using the WAF scheme, the HLLC and Roe schemes can be expressed in similar form. The HLLC and Roe schemes are tested against Quirk's test problems, and shock instability appears in both schemes. To remedy shock instability, we propose a control method of flux difference across the contact and shear waves. To catch shock waves, an appropriate pressure sensing function is defined. Using the proposed method, shock instabilities are successfully controlled. For the Roe scheme, a modified Harten-Hyman entropy fix method using Harten-Lax-van Leer-type switching is suggested. A suitable criterion for switching is established, and the modified Roe scheme works successfully with the suggested method.