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A Specification of a Control System for Adaptive Time Stepping

Söderlind, Gustaf LU (2005)
Abstract
To increase their computational efficiency, time-stepping methods for the solution of ODEs and DAEs are usually adaptive. A control system for the adaptivity includes the automatic control of step size as well as method order, and can be constructed using proven design principles from control theory. It is a ''superstructure,'' surrounding the basic method, that manages the computational process.



As the control system is a dynamical system in its own right, its properties need to be properly analyzed and tested, motivating a separate study of control system dynamics. Here we propose a complete specification of a control system for adaptive time-stepping, based on controlling local error estimates. The control system is... (More)
To increase their computational efficiency, time-stepping methods for the solution of ODEs and DAEs are usually adaptive. A control system for the adaptivity includes the automatic control of step size as well as method order, and can be constructed using proven design principles from control theory. It is a ''superstructure,'' surrounding the basic method, that manages the computational process.



As the control system is a dynamical system in its own right, its properties need to be properly analyzed and tested, motivating a separate study of control system dynamics. Here we propose a complete specification of a control system for adaptive time-stepping, based on controlling local error estimates. The control system is specified in Matlab to facilitate ease of implementation and is executable on its own. The entire control system is independent of the basic discretization method, and can be tested stand--alone, without any specific ODE or DAE method.



With respect to step size control, a collection consisting of PI controllers and digital filters is included. For

controlling the order, error estimates for two alternative methods of different orders are monitored. The estimates are fed to the same step size controller, which will produce different step size suggestions to estimate relative efficiency. The order change is based on simple time integration of the efficiency measure. Anti-windup is used to counteract excessive step size increases for method orders currently not in use. We further discuss the choice and handling of the norm used for measuring local errors. (Less)
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organization
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type
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publication status
submitted
subject
keywords
adaptivity control theory digital filters error control stepsize control
publisher
[Publisher information missing]
language
English
LU publication?
yes
id
acd20d70-4e55-45f7-aa3c-925729671467 (old id 633954)
date added to LUP
2008-01-11 14:47:03
date last changed
2017-02-13 13:10:04
@techreport{acd20d70-4e55-45f7-aa3c-925729671467,
  abstract     = {To increase their computational efficiency, time-stepping methods for the solution of ODEs and DAEs are usually adaptive. A control system for the adaptivity includes the automatic control of step size as well as method order, and can be constructed using proven design principles from control theory. It is a ''superstructure,'' surrounding the basic method, that manages the computational process.<br/><br>
<br/><br>
As the control system is a dynamical system in its own right, its properties need to be properly analyzed and tested, motivating a separate study of control system dynamics. Here we propose a complete specification of a control system for adaptive time-stepping, based on controlling local error estimates. The control system is specified in Matlab to facilitate ease of implementation and is executable on its own. The entire control system is independent of the basic discretization method, and can be tested stand--alone, without any specific ODE or DAE method.<br/><br>
<br/><br>
With respect to step size control, a collection consisting of PI controllers and digital filters is included. For<br/><br>
controlling the order, error estimates for two alternative methods of different orders are monitored. The estimates are fed to the same step size controller, which will produce different step size suggestions to estimate relative efficiency. The order change is based on simple time integration of the efficiency measure. Anti-windup is used to counteract excessive step size increases for method orders currently not in use. We further discuss the choice and handling of the norm used for measuring local errors.},
  author       = {Söderlind, Gustaf},
  institution  = {[Publisher information missing]},
  keyword      = {adaptivity
control theory
digital filters
error control
stepsize control},
  language     = {eng},
  title        = {A Specification of a Control System for Adaptive Time Stepping},
  year         = {2005},
}