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