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In this paper we propose a novel way to integrate time-evolving partialdifferential equations that contain nonlinear advection and stiff linearoperators, combining exponential integration techniques and semi-Lagrangianmethods.  The general formulation is built from the solution of an integration factorproblem with respect to the problem written with a material derivative, so thatthe exponential integration scheme naturally incorporates the nonlinearadvection. Semi-Lagrangian techniques are used to treat the dependence of theexponential integrator on the flow trajectories. The formulation is general, asmany exponential integration techniques could be combined with differentsemi-Lagrangian methods. This formulation allows an accurate solution of thelinear stiff operator, a property inherited by the exponential integrationtechnique. It also provides a sufficiently accurate representation of thenonlinear advection, even with large time-step sizes, a property inherited bythe semi-Lagrangian method.  Aiming for application in weather and climate modeling, we discuss possiblecombinations of well established exponential integration techniques andstate-of-the-art semi-Lagrangian methods used operationally in the application.We show experiments for the planar rotating shallow water equations. Whencompared to traditional exponential integration techniques, the experimentsreveal that the coupling with semi-Lagrangian allows stabler integration withlarger time-step sizes. From the application perspective, which already usessemi-Lagrangian methods, the exponential treatment could improve the solutionof wave-dispersion when compared to semi-implicit schemes.

Semi-Lagrangian Exponential Integration with application to the rotating shallow water equations