Summation-by-Parts Finite-Difference Method for Linear Shallow Water Equations on Staggered Curvilinear Grids in Closed Domains

TL;DR

This paper develops high-order, energy-stable staggered finite-difference schemes for linear shallow water equations on complex curvilinear grids, improving wave simulation accuracy in ocean and atmospheric models.

Contribution

It extends SBP FD methods to non-orthogonal curvilinear grids and introduces new interface treatment techniques for enhanced stability and accuracy.

Findings

Significant accuracy improvements over collocated methods

Effective handling of interface conditions on complex grids

Enhanced wave dynamics capture in numerical experiments

Abstract

This work focuses on developing high-order energy-stable schemes for wave-dominated problems in closed domains using staggered finite-difference summation-by-parts (SBP FD) operators. We extend the previously presented uniform staggered grid SBP FD approach to non-orthogonal curvilinear multi-block grids and derive new higher-order approximations. The combination of Simultaneous-Approximation-Terms (SAT) and projection method is proposed for the treatment of interface conditions on a staggered grid. This reduces approximation stiffness and mitigates stationary wave modes of pure SAT approach. Also, energy-neutral discrete Coriolis terms operators are presented. The proposed approach is tested using the linearized shallow water equations on a rotating sphere, a testbed relevant for ocean and atmospheric dynamics. Numerical experiments show significant improvements in capturing wave dynamics compared to collocated SBP FD methods.