This study proposes a new formulation for the Harten, Lax, and van Leer (HLL) type Riemann solver which is capable of solving contact discontinuities accurately but with robustness for strong shock. It is well known that the original HLL, which has incomplete wave structures, is too dissipative to capture contact discontinuities accurately. On the other side, contact-capturing approximate Riemann solvers such as HLL with Contact (HLLC) usually suffer from spurious solutions, also called carbuncle phenomenon, for strong shock. In this work, a new accurate and robust HLL-type formulation, the so-called HLL-BVD (HLL Riemann solver with boundary variation diminishing) is proposed by modifying the original HLL with the BVD algorithm. Instead of explicitly recovering the complete wave structures like the way of HLLC, the proposed method restores the missing contact with a jump-like function. The capability of solving contact discontinuities is further improved by minimizing the inherent dissipation term in HLL. Without modifying the original incomplete wave structures of HLL, the robustness for strong shock has been reserved. Thus, the proposed method is free from the shock instability problem. The accuracy and robustness of the new method are demonstrated through solving several one- and two-dimensional tests. Results indicate that the new formulation based on the two-wave HLL-type Riemann solver is not only capable of capturing contact waves more accurately than the original HLL or HLLC but, most importantly, is free form carbuncle instability for strong shock.
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