Splitting methods with complex times for parabolic equations
François Castella, Philippe Chartier, Stéphane Descombes and Gilles Vilmart
Abstract.
Using composition procedures, we build up high order splitting methods
to solve evolution equations posed in finite or infinite dimensional spaces.
Since high-order splitting methods with real time are known to involve
large and/or negative time steps, which destabilizes the overall procedure,
the key point of our analysis is, we develop splitting methods that use
complex time steps having positive real part:
going to the complex plane allows to considerably
increase the accuracy, while keeping small time steps; on the other hand,
restricting our attention to time steps with
positive real part makes our methods more stable,
and in particular well adapted in the case when the considered evolution equation
involves unbounded operators in infinite dimensional spaces, like parabolic (diffusion) equations.
We provide a thorough analysis in the case of linear equations posed in
general Banach spaces. We also numerically investigate the nonlinear
situation. We illustrate our results in the case of (linear and nonlinear) parabolic equations.