Adaptive time stepping shows implicit methods often unnecessary for PDEs
Are implicit schemes needed in adaptive time stepping for PDEs?
Computational Engineering, Finance, and Science
Summary
When solving changing physical problems using equations on a computer, people often choose special methods called implicit schemes to avoid taking tiny steps in time. But sometimes, the step size has to be small anyway to keep the answer accurate. The authors tested whether these special methods are really needed or if simpler, faster methods can work just as well. They found that in certain situations, simpler methods are almost as good, which can save a lot of computing work.
What this means in practice
- •For scientific computing teams: Choose simpler time integration methods to reduce computational costs while maintaining accuracy in time-adaptive PDE simulations.
- •For high-performance computing engineers: Design large-scale PDE solvers optimized for parallel systems by replacing implicit schemes with stabilized explicit ones when adaptive accuracy bounds time steps.
Authors
Mikhail A. Botchev
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.