Analysis and improvement of a semi-Lagrangian exponential scheme for 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 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.
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