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Multi-adaptive time integration

Journal article
Authors Anders Logg
Published in Applied Numerical Mathematics
Volume 48
Issue 3-4
Pages 339-354
ISSN 0168-9274
Publication year 2004
Published at Department of Computational Mathematics
Pages 339-354
Language en
Keywords Adaptivity , Error control , Explicit , Multi-adaptivity , Stiffness
Subject categories Mathematics, Computational Mathematics

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. © 2003 IMACS. Publised by Elsevier B.V. All rights reserved.

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