Time-step selection algorithms: Adaptivity, control, and signal processing

Authors:
Gustaf Söderlind
Affiliations:
Numerical Analysis, Centre for Mathematical Sciences, Lund University, Box 118, S-221 00 Lund, Sweden
Venue:
Applied Numerical Mathematics
Year:
2006

Citing 12
Cited 4

Equilibrium states of Runge-Kutta schemes: part II

ACM Transactions on Mathematical Software (TOMS)
A PI stepsize control for the numerical solution of ordinary differential equations

BIT
Embedded Runge-Kutta formulae with stable equilibrium states

Journal of Computational and Applied Mathematics
Control-theoretic techniques for stepsize selection in implicit Runge-Kutta methods

ACM Transactions on Mathematical Software (TOMS)
Control theoretic techniques for stepsize selection in explicit Runge-Kutta methods

ACM Transactions on Mathematical Software (TOMS)
Equilibrium states of Runge Kutta schemes

ACM Transactions on Mathematical Software (TOMS)
Control Strategies for the Iterative Solution of Nonlinear Equations in ODE Solvers

SIAM Journal on Scientific Computing
On the construction of error estimators for implicit Runge-Kutta methods

Journal of Computational and Applied Mathematics
Stage Value Predictors and Efficient Newton Iterations in Implicit Runge--Kutta Methods

SIAM Journal on Scientific Computing
Numerical Initial Value Problems in Ordinary Differential Equations

Numerical Initial Value Problems in Ordinary Differential Equations
Digital filters in adaptive time-stepping

ACM Transactions on Mathematical Software (TOMS)
Explicit, Time Reversible, Adaptive Step Size Control

SIAM Journal on Scientific Computing

Doubly quasi-consistent parallel explicit peer methods with built-in global error estimation

Journal of Computational and Applied Mathematics
Variable-Stepsize Interpolating Explicit Parallel Peer Methods with Inherent Global Error Control

SIAM Journal on Scientific Computing
Embedded exponential operator splitting methods for the time integration of nonlinear evolution equations

Applied Numerical Mathematics
A stepsize control algorithm for SDEs with small noise based on stochastic Runge---Kutta Maruyama methods

Numerical Algorithms

Quantified Score

Hi-index	0.00

Visualization

Abstract

The efficiency of numerical time-stepping methods for dynamical systems is greatly enhanced by automatic time step variation. In this paper we present and discuss three different approaches to step size selection: (i) control theory (to keep the error in check); (ii) signal processing (to produce smooth step size sequences and improve computational stability); and (iii) adaptivity, in the sense that the time step should be covariant or contravariant with some prescribed function of the dynamical system's solution. Examples are used to demonstrate the different advantages in different applications. The main ideas are further developed to approach some open problems that are subject to special requirements.