Spectral interpolation in semi-implicit semi-Lagrangian methods for shallow water equations on the sphere

TL;DR

This paper introduces a spectrally accurate interpolation method for semi-implicit semi-Lagrangian schemes solving the shallow water equations on the sphere, improving accuracy and conservation over traditional low-order methods.

Contribution

The work is the first to implement and evaluate spectrally accurate interpolation in SISL schemes for the SWE, leveraging the DFS model and NUFFT for efficiency.

Findings

Higher accuracy in test cases compared to low-order interpolation

Improved mass and energy conservation

Reduced numerical diffusion over long simulations

Abstract

Semi-implicit semi-Lagrangian (SISL) methods are commonly used for the shallow water equations (SWE) because they allow for larger time steps than those permitted by the Courant-Friedrichs-Lewy (CFL) stability condition in Eulerian schemes. In these methods, the semi-Lagrangian treatment of advection is typically performed using lower-order interpolation, such as tensor-product Lagrange interpolation with cubic or quintic polynomials. However, operational SISL schemes routinely employ spectrally accurate spatial discretizations, such as spherical harmonics or the double Fourier sphere (DFS) method, for computing horizontal derivatives of the prognostic variables. This creates a mismatch in numerical accuracy, making the use of low-order interpolation less clearly justified. In this work, we present the first numerical investigation of spectrally accurate interpolation in SISL schemes for the SWE. Our approach builds upon the recently developed DFS-based SWE model, incorporating a spectral interpolation scheme that is accelerated using the nonuniform fast Fourier transform (NUFFT) to maintain the same overall computational complexity as the original model. Using several standard SWE test cases, we evaluate the accuracy, conservation, and numerical diffusion of the new model, particularly over long integration times. Compared to an equivalent SISL model with low-order interpolation, the new model achieves higher accuracy, improved mass and energy conservation, and reduced numerical diffusion, demonstrating the potential benefits of incorporating spectrally accurate interpolation into SISL schemes.