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

TL;DR

The paper develops high-order energy-stable SBP-FD schemes on staggered grids for closed-domain wave problems, extending to non-orthogonal curvilinear multi-block grids and introducing a hybrid SAT-projection interface treatment with energy-neutral Coriolis terms. It focuses on the linearized shallow water equations on a rotating sphere to assess dispersion, stability, and geophysical-wave dynamics, showing significant improvements over collocated SBP-FD methods in gravity-wave propagation and Rossby-adjusted features. The key contributions are the 6th-order staggered SBP-FD operators on curvilinear multi-block grids, the SAT-projection interface approach that eliminates spurious fast/stationary modes, and energy-conserving Coriolis discretizations that preserve mimetic properties like mass and energy conservation. The results have practical impact for ocean-atmosphere modeling, enabling larger time steps and more accurate wave dynamics on complex grids, with applications to geophysical fluid dynamics and related wave-dominated systems.

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.

Figures (11)

• Figure 1: Eigenvalues of the discrete Laplace scaled by $\Delta x^2$ using SAT (red circles) or SAT-projection method (blue pluses) to impose interface conditions.
• Figure 2: Arrangement of blocks and grid points in each block of equiangular gnomonic cubed-sphere grid.
• Figure 3: Dependence of 25-days simulation maximum values of $l_2$ and $l_\infty$ error norms for numerical solution of Gaussian hill test variant 1.
• Figure 4: As Fig. \ref{['fig:gauss_conv']}, but for test variant 2.
• Figure 5: As Fig. \ref{['fig:gauss_conv']}, but for test variant 3.

Theorems & Definitions (2)

• Remark 2.1
• Remark 2.2