Analysis and improvement of a semi-Lagrangian exponential scheme for the shallow-water equations on the rotating sphere
<|reference_start|>Analysis and improvement of a semi-Lagrangian exponential scheme for the shallow-water equations on the rotating sphere: 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 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.<|reference_end|>
@article{steinstraesser2024analysis,
    title={Analysis and improvement of a semi-Lagrangian exponential scheme for the
  shallow-water equations on the rotating sphere},
    author={Jo~ao Guilherme Caldas Steinstraesser, Martin Schreiber, Pedro da
  Silva Peixoto},
    journal={arXiv preprint arXiv:2405.02237},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.02237},
    primaryClass={math.NA cs.NA}
}