Multi-Adaptive Time-Integration

TL;DR

This paper discusses multi-adaptive Galerkin methods for efficient time integration of systems with multiple time scales, introducing an adaptive stabilization strategy for explicit methods applied to stiff ODEs.

Contribution

It presents a novel adaptive stabilization approach for explicit multi-adaptive methods to handle stiffness in ODEs effectively.

Findings

Multi-adaptive methods adapt time steps per component based on error estimates.

The proposed stabilization strategy improves explicit method convergence for stiff systems.

Numerical experiments demonstrate enhanced efficiency and stability.

Abstract

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.