| Issue |
ESAIM: M2AN
Volume 59, Number 3, May-June 2025
|
|
|---|---|---|
| Page(s) | 1531 - 1564 | |
| DOI | https://doi.org/10.1051/m2an/2025034 | |
| Published online | 04 June 2025 | |
Analysis and improvement of a semi-Lagrangian exponential scheme for the shallow-water equations on the rotating sphere
1
Applied Mathematics, Universidade de S˜ao Paulo, S˜ao Paulo, CEP 05508-090, Brazil
2
Université Grenoble Alpes/Laboratoire Jean Kuntzmann, Saint-Martin-d’Heres 38400, France
3
Inria AIRSEA Team, Grenoble 38058, France
4
Technical University of Munich, Garching 85748, Germany
* Corresponding author: joao.steinstraesser@usp.br; joao.steinstraesser@gmail.com
Received:
1
July
2024
Accepted:
22
April
2025
In this work, we study and extend a class of semi-Lagrangian exponential methods, which combine exponential time integration techniques, suitable for integrating stiff linear terms, with a semi-Lagrangian treatment of nonlinear advection terms. Partial differential equations involving both processes arise for instance in atmospheric circulation models. Through a truncation error analysis, we show that previously formulated semi-Lagrangian exponential schemes are limited to first-order accuracy due to the approximation of the integration factor acting on the discretization of the linear term; we then formulate a new discretization leading to second-order accuracy. Also, a detailed stability study is conducted to compare several Eulerian and semi-Lagrangian exponential schemes, as well as a well-established semi-Lagrangian semi-implicit method, which is used in operational atmospheric models. Numerical simulations of the shallow-water equations on the rotating sphere are performed to assess the orders of convergence, stability properties, and computational cost of each method. The proposed second-order semi-Lagrangian exponential method was shown to be more stable and accurate than the previously formulated schemes of the same class at the expense of larger wall-clock times; however, the method is more stable and has a similar cost compared to the well-established semi-Lagrangian semi-implicit method; therefore, it is a competitive candidate for potential operational applications in atmospheric circulation modeling.
Mathematics Subject Classification: 65M12 / 65M22 / 76U60
Key words: Exponential integrator / semi-Lagrangian / accuracy and stability analysis / shallow-water equations on the rotating sphere / atmospheric modeling
© The authors. Published by EDP Sciences, SMAI 2025
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