Time Parallelization for Nonlinear Problems Based on Diagonalization

摘要

We have introduced a new method which allows us to use diagonalization for time parallelization also for nonlinear problems. We have shown two variants for nonlinear scalar problems, and one for a nonlinear PDE. Numerical experiments show that the methods converge rapidly, and there is also an optimal choice of the geometric time grid stretching, like in the original algorithm for linear problems (Gander et al. 2014, 2016a). The geometric stretching is only one way to make diagonalization possible: random or adaptive time steps could also be used, but they must be determined for the entire time window before its parallel solve, and they must all be different, otherwise the diagonalization is not possible. In an adaptive setting, one could adaptively determine a macro time step with a larger tolerance as time window, before parallelizing its solve with smaller geometric or random time steps. We are currently investigating such variants, and also the generalization to nonlinear hyperbolic problems.