Strongly stable dual-pairing summation by parts finite difference schemes for the vector invariant nonlinear shallow water equations – I: Numerical scheme and validation on the plane

Abstract

We present an energy/entropy stable and high order accurate finite difference (FD) method for solving the nonlinear (rotating) shallow water equations (SWEs) in vector invariant form using the newly developed dual-pairing and dispersion-relation preserving summation by parts (SBP) FD operators. We derive new well-posed boundary conditions (BCs) for the SWE in one space dimension, formulated in terms of fluxes and applicable to linear and nonlinear SWEs. For the nonlinear vector invariant SWE in the subcritical regime, where energy is an entropy functional, we find that energy/entropy stability ensures the boundedness of numerical solution but does not guarantee convergence. Adequate amount of numerical dissipation is necessary to control high frequency errors which could negatively impact accuracy in the 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 minimises oscillations from shocks and discontinuities, and eliminates high frequency grid-scale errors. The numerical method is most suitable for the simulations of subcritical flows typically observed in atmospheric and geostrophic flow problems. We prove both nonlinear and local linear stability results, as well as a priori error estimates for the semi-discrete approximations of both linear and nonlinear SWEs. Convergence, accuracy, and well-balanced properties are verified via the method of manufactured solutions and canonical test problems such as the dam break and lake at rest. Numerical simulations in two-dimensions are presented which include the rotating and merging vortex problem and barotropic shear instability, with fully developed turbulence.