Development of a Hermite-like weighted compact nonlinear scheme based on the two-stage fourth-order temporal accurate framework

We develop in this work a fifth-order Hermite-like weighted compact nonlinear scheme (HL-WCNS) within the framework of a two-stage fourth-order Lax–Wendroff type time discretization [Li and Du, “A two-stage fourth order time-accurate discretization for Lax–Wendroff type flow solvers I. Hyperbolic conservation laws,” SIAM J. Sci. Comput. 38, A3046–A3069 (2016)] based on the generalized Riemann problem (GRP) solver for hyperbolic conservation laws in this paper. While in the conventional WCNS, only the nodal values are solved, the HL-WCNS also evolves the midpoint values using a two-stage fourth-order temporal accurate scheme. The midpoint flow variables and the nodal values are obtained using the information particularly provided by the GRP solver. The obtained midpoint values approximate the first derivatives at the nodes. Then, a high-order nonlinear interpolation is proposed using the nodal values and their derivatives over a more compact stencil. There is no additional effort to compute the first derivative of the flow variable as in the standard Hermite weighted essentially nonoscillatory (WENO) schemes or discontinuous Galerkin (DG) approach. An efficient limiting technique to enforce the positivity-preserving property is proposed in conjunction with the proposed scheme, specifically designed to address strong discontinuities. Owing to its compact stencils in space and time, this new scheme has been demonstrated with high numerical resolution through benchmark test cases.