Analysis and improvement of a semi-Lagrangian exponential scheme for the shallow-water equations on the rotating sphere
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| Format: | Preprint |
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2024
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| _version_ | 1866917996571656192 |
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| author | Steinstraesser, João Guilherme Caldas Schreiber, Martin Peixoto, Pedro da Silva |
| author_facet | Steinstraesser, João Guilherme Caldas Schreiber, Martin Peixoto, Pedro da Silva |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_02237 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Analysis and improvement of a semi-Lagrangian exponential scheme for the shallow-water equations on the rotating sphere Steinstraesser, João Guilherme Caldas Schreiber, Martin Peixoto, Pedro da Silva Numerical Analysis 65M12, 65M22, 76U60 G.1.8; G.1.10 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. |
| title | Analysis and improvement of a semi-Lagrangian exponential scheme for the shallow-water equations on the rotating sphere |
| topic | Numerical Analysis 65M12, 65M22, 76U60 G.1.8; G.1.10 |
| url | https://arxiv.org/abs/2405.02237 |