The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods
The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods
Adaptive finite element methods for parabolic problems. I.: a linear model problem
SIAM Journal on Numerical Analysis
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
SIAM Journal on Numerical Analysis
Adaptive finite element methods for parabolic problems II: optimal error estimates in L∞L2 and L∞L∞
SIAM Journal on Numerical Analysis
Adaptive finite element methods for parabolic problems IV: nonlinear problems
SIAM Journal on Numerical Analysis
Adaptive finite element methods for parabolic problems V: long-time integration
SIAM Journal on Numerical Analysis
Explicit Runge-Kutta methods for parabolic partial differential equations
Applied Numerical Mathematics - Special issue celebrating the centenary of Runge-Kutta methods
Journal of Parallel and Distributed Computing - Special issue on dynamic load balancing
Adaptive Finite Element Methods for Parabolic Problems VI: Analytic Semigroups
SIAM Journal on Numerical Analysis
High Resolution Schemes for Conservation Laws with Locally Varying Time Steps
SIAM Journal on Scientific Computing
Projective Methods for Stiff Differential Equations: Problems with Gaps in Their Eigenvalue Spectrum
SIAM Journal on Scientific Computing
Multi-Adaptive Galerkin Methods for ODEs I
SIAM Journal on Scientific Computing
Explicit Time-Stepping for Stiff ODEs
SIAM Journal on Scientific Computing
Multi-Adaptive Galerkin Methods for ODEs II: implementation and Applications
SIAM Journal on Scientific Computing
A multirate time integrator for regularized Stokeslets
Journal of Computational Physics
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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 meG(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.