Analysis and improvement of a semi-Lagrangian exponential scheme for the shallow-water equations on the rotating sphere

TL;DR

This paper develops a second-order semi-Lagrangian exponential scheme for the shallow-water equations on the rotating sphere, improving accuracy and stability over previous methods and comparing favorably with existing semi-implicit schemes for atmospheric modeling.

Contribution

The paper introduces a second-order discretization for semi-Lagrangian exponential schemes, enhancing accuracy and stability for atmospheric equations on the rotating sphere.

Findings

Second-order scheme is more stable and accurate than first-order versions.

The new method has similar computational cost to semi-Lagrangian semi-implicit schemes.

Numerical results confirm improved convergence and stability properties.

Abstract

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.