An artificially upstream flux vector splitting scheme for the Euler equations

A new method is proposed to split the flux vector of the Euler equations by introducing two artificial wave speeds. The direction of wave propagation is adjusted by these two wave speeds. If they are set to be the fastest wave speeds in two opposite directions, the method leads to the HLL approximate Riemann solver devised by Harten, Lax and van Leer, which indicates that the HLL solver is a vector flux splitting scheme as well as a Godunov-type scheme. A more accurate scheme that resolves 1D contact discontinuity is further proposed by carefully choosing two wave speeds so that the flux vector is split to two simple flux vectors. One flux vector comes with either non-negative or non-positive eigenvalues and is 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 1D Euler equations, the scheme actually surpasses the exact solver in avoiding expansion shocks without any additional entropy fix. The scheme can exactly resolve stationary 1D contact discontinuities, and it avoids the carbuncle problem in multi-dimensional computations. The robustness of the scheme is shown in 1D test cases designed by Toro, and other 2D calculations.