  In this paper we propose a novel way to integrate time-evolving partial
differential equations that contain nonlinear advection and stiff linear
operators, combining exponential integration techniques and semi-Lagrangian
methods.
  The general formulation is built from the solution of an integration factor
problem with respect to the problem written with a material derivative, so that
the exponential integration scheme naturally incorporates the nonlinear
advection. Semi-Lagrangian techniques are used to treat the dependence of the
exponential integrator on the flow trajectories. The formulation is general, as
many exponential integration techniques could be combined with different
semi-Lagrangian methods. This formulation allows an accurate solution of the
linear stiff operator, a property inherited by the exponential integration
technique. It also provides a sufficiently accurate representation of the
nonlinear advection, even with large time-step sizes, a property inherited by
the semi-Lagrangian method.
  Aiming for application in weather and climate modeling, we discuss possible
combinations of well established exponential integration techniques and
state-of-the-art semi-Lagrangian methods used operationally in the application.
We show experiments for the planar rotating shallow water equations. When
compared to traditional exponential integration techniques, the experiments
reveal that the coupling with semi-Lagrangian allows stabler integration with
larger time-step sizes. From the application perspective, which already uses
semi-Lagrangian methods, the exponential treatment could improve the solution
of wave-dispersion when compared to semi-implicit schemes.
