Fully discrete WENO with double entropy condition for hyperbolic conservation laws

Abstract

ABSTRACTThis paper put forward a new fully discrete scheme construction method – double entropy condition solution formula method. With that, we turn the state-of-the-art semi-discrete WENO + RK scheme into a fully discrete scheme, which is named as Full-WENO. A major difficulty of this work is that we lack exact solution expressions for nonlinear equations in general cases. A feasible way we can go is to linearize equations and get quasi-exact solution formulas. The critical challenge is keeping both accuracy and efficiency in a scheme. Then, we get a class of new high-order schemes far better than traditional WENO schemes in the following aspects: (1) One-step to consistent high accuracy order in both space and time; (2) Resolution improves with the increasing CFL number; (3) Less CPU time and memory space, 1/s times of WENO with s-stage RK method in theory; (4) Excellent entropy condition satisfying property. Compared with our original work , the new method applies the more sophisticated WENO reconstruction and solves the resolution loss problems in multi-dimensional cases. The numerical tests show that the new scheme is equipped with the merits of high efficiency, high resolution and low dissipation, especially for long-time nonlinear problems.