Time integration by operator splitting

Alexander Ostermann, University of Innsbruck
October 9th, 2013 at 3:30PM–4:30PM in 939 Evans Hall [Map]

Splitting methods form an important class of competitive time integration schemes for evolution equations. Despite of their extensive use in real-life applications, splitting methods for partial differential equations are still far from being fully understood. In this talk, we will discuss some rigorous results that were obtained recently.

First, a convergence analysis of the Strang splitting algorithm with a discontinuous Galerkin approximation in space for the Vlasov–Poisson equations is provided. The Vlasov equation models the behavior of a collisionless plasma in astro- and plasma physics. In a simplified model, the force term is given by the gradient of the self-consistent electric potential

As a second example, splitting schemes for reaction-diffusion equations are discussed. Splitting the nonlinear reaction terms from the linear diffusion operator gives rise to attractive numerical schemes: the semiflow generated by the diffusion operator can be obtained by standard fast solvers (relying on fast Fourier transform techniques or fast Poisson solvers); the nonlinear part being local only requires the solution of an ordinary differential equation. Moreover, such splitting methods will preserve the positivity of the solution in certain situations.