A Second-Order Semi-Implicit Projection Time-Stepping Scheme for the Shallow-Water Equations on the Sphere Using Compatible Finite Elements
Abstract
We propose a second-order, semi-implicit time-stepping scheme for the shallow-water equations on the sphere discretised with compatible finite elements. The scheme splits the advection operator from the wave operator, applying a DG-upwind SSPRK3 method to the advection part and an H(div)-conforming implicit discretisation to the wave part, solved efficiently via hybridisation. The scheme follows a three-step symmetric splitting: an explicit half-step, a full advection step, and an implicit half-step that simultaneously projects the velocity back into the H(div) space. This symmetry guarantees global second-order accuracy and confines the stability restriction to the CFL condition of the advection step alone. The time-step can therefore be extended beyond this limit by subcycling the advection step without additional implicit solves. The method is implemented in Firedrake and validated against a standard suite of spherical shallow-water benchmarks, demonstrating stable results for Courant numbers up to 1 (with two advection subcycles) with second-order finite elements, second-order convergence, and good parallel scaling.
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BibTeX
@misc{kowalczyk2026secondorder,
title = {{A Second-Order Semi-Implicit Projection Time-Stepping Scheme for the Shallow-Water Equations on the Sphere Using Compatible Finite Elements}},
author = {Karina Kowalczyk and Colin Cotter},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08169},
eprint = {2611.08169},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08169},
}
RIS
TY - UNPB
TI - A Second-Order Semi-Implicit Projection Time-Stepping Scheme for the Shallow-Water Equations on the Sphere Using Compatible Finite Elements
AU - Karina Kowalczyk
AU - Colin Cotter
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08169
UR - https://arxiv.org/abs/2611.08169
AB - We propose a second-order, semi-implicit time-stepping scheme for the shallow-water equations on the sphere discretised with compatible finite elements. The scheme splits the advection operator from the wave operator, applying a DG-upwind SSPRK3 method to the advection part and an H(div)-conforming implicit discretisation to the wave part, solved efficiently via hybridisation. The scheme follows a three-step symmetric splitting: an explicit half-step, a full advection step, and an implicit half-step that simultaneously projects the velocity back into the H(div) space. This symmetry guarantees global second-order accuracy and confines the stability restriction to the CFL condition of the advection step alone. The time-step can therefore be extended beyond this limit by subcycling the advection step without additional implicit solves. The method is implemented in Firedrake and validated against a standard suite of spherical shallow-water benchmarks, demonstrating stable results for Courant numbers up to 1 (with two advection subcycles) with second-order finite elements, second-order convergence, and good parallel scaling.
ER -