Construction of accurate, robust and simple riemann solvers

A new method is proposed to split the flux vector of the Euler equations by introducing two artificial wave speeds. The flux vector is split to two simple ones. One flux vector comes with unidirectional eigenvalues, so that it can be easily solved by one-side differencing. Another flux vector becomes a system of two waves and one, two or three stationary discontinuities depending on the dimension of the Euler equations. Numerical flux function for multi-dimensional Euler equations is formulated for any grid system, structured or unstructured. A remarkable simplicity of the scheme is that it successfully achieves one-sided approximation for all waves without recourse to any matrix operation. Moreover, its accuracy is comparable with the exact Riemann solver. For 1-D Euler equations, the scheme actually surpasses the exact solver in avoiding expansion shocks without any additional entropy fix. The scheme can exactly resolve stationary 1-D contact discontinuities, and it avoids the carbuncle problem in multi-dimensional computations. It is shear diffusion terms that suppress the carbuncle problem. It is found that the shear diffusion terms can be used to suppress the carbuncle problem produced by other upwind schemes as well, such as the Godunov and the HLLC Riemann solvers. Side effect of the shear dissipation in the computation of the boundary layer can be controlled within a low level if the limiting procedure for a high-order scheme is not imposed on tangential velocity.