Improvements for matrix-type implicit temporal scheme regarding entropy fix and local time step

The matrix version of Symmetric Successive Over Relaxation (matrix-SSOR) scheme has been proved to be more efficient than the standard Lower-Upper Symmetric Gauss-Seidel (LU-SGS), but less robust for high-speed flows. In order to ulteriorly improve the convergence rate as well as numerical stability of matrix-SSOR, two improvements regarding entropy fix and local time step have been proposed and validated. Firstly, an augmented entropy fix method is imposed on the inviscid Jacobian matrix and proved to be effective in two high-speed flows, in which the key parameter in entropy fix is discussed and found to be insensitive within appropriate range of values. Since the time step also has great effects on the numerical stability and convergence rate, a modified cell residual adapted local time step method with consideration of the residual history is developed, which is found to be effective for increasing the convergence rate when the matrix-SSOR is applied, but invalid when the LU-SGS is used. The proposed modified local time step method is also insensitive to the key parameter within appropriate range of values. The two modifications can be conveniently implanted into analogous matrix-type implicit schemes to improve the numerical performance.