Are implicit schemes needed in adaptive time stepping for PDEs?

Abstract

Implicit time integration is the standard way to overcome the tight CFL time step restriction when solving nonstationary PDEs numerically. However, within an adaptive time accurate integration, the time step size selection is governed not only by stability but also by accuracy requirements. Therefore, an eligible question to ask is: is (unconditionally) stable implicit time stepping really needed if the time step is restricted by accuracy considerations anyway? And how much do we then lose (in terms of the time step size) when switching from the expensive implicit time stepping to stabilized explicit schemes? An answer to this question is also important because such a switch is often done when solving large scale nonstationary PDEs on massively parallel computers. This paper attempts to answer these questions by numerical tests for two particular adaptive time accurate settings: (i)~nonstationary convection-diffusion equations and (ii)~nonlinear heat equations.