Control of error and convergence in ODE solvers
Gustafsson, K
99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; DATA COVARIANCES; CONTROL THEORY; DIFFERENTIAL EQUATIONS; ALGORITHMS; COMPUTER CODES; COMPUTERIZED SIMULATION; CONVERGENCE; EXPERIMENTAL DATA; INTEGRAL CALCULUS; RUNGE-KUTTA METHOD; 990200; MATHEMATICS AND COMPUTERS
Feedback is a general principle that can be used in many different contexts. In this thesis it is applied to numerical integration of ordinary differential equations. An advanced integration method includes parameters and variables that should be adjusted during the execution. In addition, the integration method should be able to automatically handle situations such as: initialization, restart after failures, etc. In this thesis we regard the algorithms for parameter adjustment and supervision as a controller. The controlled measures different variable that tell the current status of the integration, and based on this information it decides how to continue. The design of the controller is vital in order to accurately and efficiently solve a large class of ordinary differential equations. The application of feedback control may appear farfetched, but numerical integration methods are in fact dynamical systems. This is often overlooked in traditional numerical analysis. We derive dynamic models that describe the behavior of the integration method as well as the standard control algorithms in use today. Using these models it is possible to analyze properties of current algorithms, and also explain some generally observed misbehaviors. Further, we use the acquired insight to derive new and improved control algorithms, both for explicit and implicit Runge-Kutta methods. In the explicit case, the new controller gives good overall performance. In particular it overcomes the problem with oscillating stepsize sequence that is often experienced when the stepsize is restricted by numerical stability. The controller for implicit methods is designed so that it tracks changes in the differential equation better than current algorithms. In addition, it includes a new strategy for the equation solver, which allows the stepsize to vary more freely. This leads to smoother error control without excessive operations on the iteration matrix. (87 refs.) (au).
Lund Univ. (Sweden). Dept. of Automatic Control
OSTI; NTIS; INIS
Sweden
1992-03-01
English
Thesis/Dissertation
Numerical Data; Technical Report
Other Information: TH: Doctoral diss. (TeknD).; PBD: Mar 1992
Medium: X; Size: [184] p.
ON: DE93612919
LUTFD2-TFRT-1036
Other: ON: DE93612919; TRN: SE9200328008017
SWDN; SCA: 990200; PA: AIX-24:008017; SN: 93000932773
2008-02-12
10119341