Multi-Adaptive Time-Integration
For researchers solving stiff ODEs/PDEs with multiple time scales, this work offers a method to reduce computational cost while maintaining accuracy, though it is an incremental extension of existing Galerkin methods.
The paper presents multi-adaptive Galerkin methods (mcG(q) and mdG(q)) that use individual, adaptive time steps for each component to efficiently integrate ODEs/PDEs with multiple time scales, and introduces an adaptive stabilization strategy for explicit integration of stiff systems.
Time integration of ODEs or time-dependent PDEs with required resolution of the fastest time scales of the system, can be very costly if the system exhibits multiple time scales of different magnitudes. If the different time scales are localised to different components, corresponding to localisation in space for a PDE, efficient time integration thus requires that we use different time steps for different components. We present an overview of the multi-adaptive Galerkin methods mcG(q) and mdG(q) recently introduced in a series of papers by the author. In these methods, the time step sequence is selected individually and adaptively for each component, based on an a posteriori error estimate of the global error. The multi-adaptive methods require the solution of large systems of nonlinear algebraic equations which are solved using explicit-type iterative solvers (fixed point iteration). If the system is stiff, these iterations may fail to converge, corresponding to the well-known fact that standard explicit methods are inefficient for stiff systems. To resolve this problem, we present an adaptive strategy for explicit time integration of stiff ODEs, in which the explicit method is adaptively stabilised by a small number of small, stabilising time steps.