Multi-Level Spectral Deferred Corrections Scheme for the Shallow Water Equations on the Rotating Sphere

TL;DR

This paper introduces a high-order implicit-explicit time integration scheme combining Multi-Level Spectral Deferred Corrections and Spherical Harmonics to efficiently solve shallow-water equations on a rotating sphere, reducing computational cost while maintaining accuracy.

Contribution

The novel MLSDC-SH scheme integrates multi-level corrections with spectral methods for improved efficiency in atmospheric flow simulations.

Findings

Achieves up to eighth-order accuracy in time.

Demonstrates significant computational speedup over single-level SDC.

Shows good stability properties in test cases.

Abstract

Efficient time integration schemes are necessary to capture the complex processes involved in atmospheric flows over long periods of time. In this work, we propose a high-order, implicit-explicit numerical scheme that combines Multi-Level Spectral Deferred Corrections (MLSDC) and the Spherical Harmonics (SH) transform to solve the wave-propagation problems arising from the shallow-water equations on the rotating sphere. The iterative temporal integration is based on a sequence of corrections distributed on coupled space-time levels to perform a significant portion of the calculations on a coarse representation of the problem and hence to reduce the time-to-solution while preserving accuracy. In our scheme, referred to as MLSDC-SH, the spatial discretization plays a key role in the efficiency of MLSDC, since the SH basis allows for consistent transfer functions between space-time levels that preserve important physical properties of the solution. We study the performance of the MLSDC-SH scheme with shallow-water test cases commonly used in numerical atmospheric modeling. We use this suite of test cases, which gradually adds more complexity to the nonlinear system of governing partial differential equations, to perform a detailed analysis of the convergence rate of MLSDC-SH upon refinement in time. We illustrate the good stability properties of MLSDC-SH and show that the proposed scheme achieves up to eighth-order accuracy in time. Finally, we study the conditions in which MLSDC-SH achieves its theoretical speedup, and we show that it can significantly reduce the computational cost compared to single-level Spectral Deferred Corrections (SDC).