Implicit-Explicit Runge-Kutta schemes and finite elements with symmetric stabilization for advection-diffusion equations

We analyze a two-stage implicit-explicit Runge–Kutta scheme for time discretization of advection-diffusion equations. Space discretization uses continuous, piecewise affine finite elements with interelement gradient jump penalty; discontinuous Galerkin methods can be considered as well. The advective and stabilization operators are treated explicitly, whereas the diffusion operator is treated implicitly. Our analysis hinges on L-energy estimates on discrete functions in physical space. Our main results are stability and quasi-optimal error estimates for smooth solutions under a standard hyperbolic CFL restriction on the time step, both in the advection-dominated and in the diffusiondominated regimes. The theory is illustrated by numerical examples. Mathematics Subject Classification. 5M12, 65M15, 65M60. Received October 28, 2010. Revised May 31, 2011. Published online February 3, 2012.