A second-order semi-Lagrangian exponential scheme with application to the shallow-water equations on the rotating sphere
arxiv(2024)
摘要
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 first 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 a second-order
accurate method. Also, a detailed stability study, both considering a linear
stability analysis and an empirical simulation-based one, 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, considering standard and challenging
benchmark test cases, 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;
therefore, it is a competitive candidate for potential operational applications
in atmospheric circulation modeling.
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