Implicit-explicit Runge-Kutta Methods with Stabilized Finite Elements for Advection-diffusion Equations
February 21,2012
3:30 p.m.
Alexandre Ern
Abstract
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 L2-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 diffusion-dominated regimes.
The theory is illustrated by numerical examples.