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In this work, we study and extend a class of semi-Lagrangian exponentialmethods, which combine exponential time integration techniques, suitable forintegrating stiff linear terms, with a semi-Lagrangian treatment of nonlinearadvection terms. Partial differential equations involving both processes arisefor instance in atmospheric circulation models. Through a truncation erroranalysis, we show that previously formulated semi-Lagrangian exponentialschemes are limited to first-order accuracy due to the approximation of theintegration factor acting on the discretization of the linear term; we thenformulate a new discretization leading to second-order accuracy. Also, adetailed stability study is conducted to compare several Eulerian andsemi-Lagrangian exponential schemes, as well as a well-establishedsemi-Lagrangian semi-implicit method, which is used in operational atmosphericmodels. Numerical simulations of the shallow-water equations on the rotatingsphere are performed to assess the orders of convergence, stability properties,and computational cost of each method. The proposed second-ordersemi-Lagrangian exponential method was shown to be more stable and accuratethan the previously formulated schemes of the same class at the expense oflarger wall-clock times; however, the method is more stable and has a similarcost compared to the well-established semi-Lagrangian semi-implicit method;therefore, it is a competitive candidate for potential operational applicationsin atmospheric circulation modeling.

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