Shock-stable roe scheme combining entropy fix and rotated riemann solver

Abstract

The Roe scheme is an important shock-capturing scheme that is known for its good performance. However, the classical Roe scheme suffers from disastrous problems, such as shock instability for hypersonic flows. The entropy fix and the rotated Riemann solver are common methods used to cure the shock instability. However, these methods may introduce a large and unnecessary numerical dissipation. To cure the shock instability with a minimally increasing numerical dissipation, this study investigates the mechanisms of entropy fix and rotated Roe scheme and combines their complementary advantages. Thus, this paper proposes the Roe-ER scheme, which uses the rotated method to identify where the entropy fix is necessary. Therefore, the possible large dissipation of the rotated Roe scheme is replaced by the relatively small value of entropy fix, and areas that adopt the entropy fix are limited in regions where the shock instability may occur. Numerical cases validate the Roe-ER scheme, which is shock stable, features a low additional numerical dissipation, and can use high-order reconstruction to further reduce areas that activate the entropy fix.