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
arxiv(2023)
摘要
We present an energy/entropy stable and high order accurate finite difference
method for solving the nonlinear (rotating) shallow water equations (SWE) in
vector invariant form using the newly developed dual-pairing and
dispersion-relation preserving summation-by-parts 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
SWEs. For the nonlinear SWE, 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
summation by parts 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 a priori error
estimates for the semi-discrete approximations of both linear and nonlinear
SWE. 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.
更多查看译文
AI 理解论文
溯源树
样例
![](https://originalfileserver.aminer.cn/sys/aminer/pubs/mrt_preview.jpeg)
生成溯源树,研究论文发展脉络
Chat Paper
正在生成论文摘要