We present an energy/entropy stable and high order accurate finite difference method for solving the linear/nonlinear shallow water equations (SWE) in vector invariant form using the newly developed dual-pairing (DP) and dispersion-relation preserving (DRP) summation by parts (SBP) finite difference operators. We derive new well-posed boundary conditions for the SWE in one space dimension, formulated in terms of fluxes and applicable to linear and nonlinear problems. For nonlinear problems, entropy stability ensures the boundedness of numerical solutions, however, it does not guarantee convergence. Adequate amount of numerical dissipation is necessary to control high frequency errors which could ruin numerical simulations. Using the dual-pairing SBP framework, we derive high order accurate and nonlinear hyper-viscosity operator which dissipates entropy and enstrophy. The hyper-viscosity operator effectively tames oscillations from shocks and discontinuities, and eliminates poisonous high frequency grid-scale errors. The numerical method is most suitable for the simulations of sub-critical flows typical observed in atmospheric and geostrophic flow problems. We prove a priori error estimates for the semi-discrete approximations of both linear and nonlinear SWE. We verify convergence, accuracy and well-balanced property via the method of manufactured solutions (MMS) and canonical test problems such as the dam break, lake at rest, and a two-dimensional rotating and merging vortex problem.

翻译：本文利用新发展的配对(DP)与频散关系保持(DRP)求和分部(SBP)有限差分算子，提出一种能量/熵稳定且高阶精度的有限差分方法，用于求解矢量不变形式下的线性/非线性浅水方程(SWE)。我们针对一维空间浅水方程推导了新的适定边界条件，该条件以通量形式表述，适用于线性和非线性问题。对于非线性问题，熵稳定性可确保数值解的有界性，但无法保证收敛性。足够的数值耗散对于控制可能破坏数值模拟的高频误差至关重要。基于配对求和分部框架，我们推导了高阶精度且非线性的超黏性算子，该算子可耗散熵与拟能。超黏性算子有效抑制激波与间断引起的振荡，并消除有害的高频网格尺度误差。该数值方法最适用于大气和地转流动问题中常见的亚临界流动模拟。我们证明了线性和非线性浅水方程半离散近似解的先验误差估计。通过制造解方法(MMS)以及经典测试问题（如溃坝、静水湖泊、二维旋转合并涡问题）验证了收敛性、精度与守恒性。