How to overcome the Courant-Friedrichs-Lewy condition of explicit discretizations?

Abstract

This manuscript contains some thoughts on the discretization of the classical heat equation. Namely, we discuss the advantages and disadvantages of explicit and implicit schemes. Then, we show how to overcome some disadvantages while preserving some advantages. However, since there is no free lunch, there is a price to pay for any improvement in the numerical scheme. This price will be thoroughly discussed below.In particular, we like explicit discretizations for the ease of their implementation even for nonlinear problems. Unfortunately, when these schemes are applied to parabolic equations, severe stability limits appear for the time step magnitude making the explicit simulations prohibitively expensive. Implicit schemes remove the stability limit, but each time step requires now the solution of linear (at best) or even nonlinear systems of equations. However, there exists a number of tricks to overcome (or at least to relax) severe stability limitations of explicit schemes without going into the trouble of fully implicit ones. The purpose of this manuscript is just to inform the readers about these alternative techniques to extend the stability limits. It was not written for classical scientific publication purposes.

Length-scales for efficient CFL conditions in high-order methods with distorted meshes: Application to local-timestepping for p-multigrid

“…The constant C is typically referred to as the "CFL-number", even though it is used in the context of both convective [6,7,8] and diffusive [9] problems. As seen above, the brunt of mispredicted h is borne by the CFL-number.…”