Fully-discrete WENO via solution formula method for hyperbolic conservation laws

Different from the spatial semi-discrete WENO schemes based on RK method in time, this paper presents a fullydiscrete scheme with consistent high order in time and space based on solution formula method. Analyzing the error composition of the framework, we obtain two core steps of discretization for constructing high-order schemes: initial value reconstruction and flux reconstruction. During the reconstruction, we apply the Newton interpolation with WENO thought and newly designed limiters to maintain robustness. Since the new method uses WENO reconstruction on fully-discrete framework, we name the new scheme as Full-WENO. With characteristics projection and Strang split technique, we extend the method to multi-dimensional Euler equations under curvilinear coordinates. The new scheme has following advantages: (1) One-step to consistent high order in time and space with excellent shock-capturing capacity; (2) Achieving exact solution in linear cases and performing better in nonlinear cases when CFL -> 1, and even can be stably implemented under CFL=1; (3) High resolution, especially for long-time evolution problems; (3) High efficiency, Full-WENO is s times faster than classical WENO with s-stage RK method under same computing condition; (4) Entropy condition automatically satisfied without additional artificial numerical viscidity. Numerical experiments contain tests of accuracy order, linear and nonlinear scalar equation, 1D and 2D Euler equations, efficiency, and sonic point. All of these verify the new scheme is equipped with the merits of high order, high resolution, and high efficiency.