A dual-pairing summation-by-parts finite difference framework for nonlinear conservation laws

Robust and stable high order numerical methods for solving partial differential equations are attractive because they are efficient on modern and next generation hardware architectures. However, the design of provably stable numerical methods for nonlinear hyperbolic conservation laws pose a significant challenge. We present the dual-pairing (DP) and upwind summation-by-parts (SBP) finite difference (FD) framework for accurate and robust numerical approximations of nonlinear conservation laws. The framework has an inbuilt "limiter" whose goal is to detect and effectively resolve regions where the solution is poorly resolved and/or discontinuities are found. The DP SBP FD operators are a dual-pair of backward and forward FD stencils, which together preserve the SBP property. In addition, the DP SBP FD operators are designed to be upwind, that is they come with some innate dissipation everywhere, as opposed to traditional SBP and collocated discontinuous Galerkin spectral element methods which can only induce dissipation through numerical fluxes acting at element interfaces. We combine the DP SBP operators together with skew-symmetric and upwind flux splitting of nonlinear hyperbolic conservation laws. Our semi-discrete approximation is provably entropy-stable for arbitrary nonlinear hyperbolic conservation laws. The framework is high order accurate, provably entropy-stable, convergent, and avoids several pitfalls of current state-of-the-art high order methods. We give specific examples using the in-viscid Burger's equation, nonlinear shallow water equations and compressible Euler equations of gas dynamics. Numerical experiments are presented to verify accuracy and demonstrate the robustness of our numerical framework.