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.

翻译：求解偏微分方程的鲁棒稳定高阶数值方法因其在现代及下一代硬件架构上的高效性而备受关注。然而，为非线性双曲守恒律设计可证明稳定的数值方法仍面临重大挑战。本文提出一种对偶配对（DP）与迎风求和分部（SBP）有限差分（FD）框架，用于实现非线性守恒律的高精度鲁棒数值逼近。该框架内置"限制器"，其目标是检测并有效处理解的分辨率不足区域和/或间断区域。DP SBP FD算子是一组后向与前向有限差分格式的对偶配对，共同保持SBP性质。此外，DP SBP FD算子被设计为迎风型，即在全域具有固有耗散特性，这与传统SBP方法和配置型间断伽辽金谱元方法形成对比——后者仅能通过单元界面处的数值通量引入耗散。我们将DP SBP算子与非线性双曲守恒律的斜对称及迎风通量分裂技术相结合。所得半离散格式对任意非线性双曲守恒律具有可证明的熵稳定性。该框架具有高阶精度、可证明的熵稳定性、收敛性，并规避了当前主流高阶方法的若干缺陷。我们以无粘性Burgers方程、非线性浅水方程和气体动力学可压缩Euler方程为例进行具体说明。数值实验验证了所提数值框架的精度并展示了其鲁棒性。