Improved WENO finite difference method: Treating the multiple discontinuities

The classical weighted essentially non-oscillatory method (WENO) performs well in solving hyperbolic conservation laws, but may encounter the numerical instability while treating multiple discontinuities due to the use of equal-width substencils. In order to overcome this problem, we propose an improved WENO finite difference method, namely WENO-rp2. The WENO-rp2 uses $r+2$ candidate substencils, which can be divided into two groups, the first group $S^1$ consists of the classical r r-point substencils and the other group $S^2$ consists of 2 2-point substencils. Then by introducing a TENO-like switching mechanism, $S^2$ is used for the final WENO-rp2 reconstruction if the classical one can not handle the multiple discontinuities or is too biased, otherwise $S^1$ is used. By doing so, the WENO-rp2 maintains the optimal $(2r-1)$th order of accuracy in the smooth region, avoids the non-physical oscillation near multiple discontinuities, and is more central, but does not significantly increase the computational cost and numerical dissipation.