Automatic Grid Control in Adaptive BVP Solvers
(2011) In Numerical Algorithms 56(1). p.6192 Abstract
 Grid adaptation in twopoint boundary value problems is usually based on mapping a uniform auxiliary grid to the desired nonuniform grid. Here we combine this approach with a new control system for constructing a grid density function I center dot(x). The local mesh width Delta x (j + 1/2) = x (j + 1) aEuro parts per thousand x (j) with 0 = x (0) < x (1) < ... < x (N) = 1 is computed as Delta x (j + 1/2) = epsilon (N) / phi (j + 1/2), where {phi j+1/2}(0) (N1) is a discrete approximation to the continuous density function I center dot(x), representing mesh width variation. The parameter epsilon (N) = 1/N controls accuracy via the choice of N. For any given grid, a solver provides an error estimate. Taking this as its input, the... (More)
 Grid adaptation in twopoint boundary value problems is usually based on mapping a uniform auxiliary grid to the desired nonuniform grid. Here we combine this approach with a new control system for constructing a grid density function I center dot(x). The local mesh width Delta x (j + 1/2) = x (j + 1) aEuro parts per thousand x (j) with 0 = x (0) < x (1) < ... < x (N) = 1 is computed as Delta x (j + 1/2) = epsilon (N) / phi (j + 1/2), where {phi j+1/2}(0) (N1) is a discrete approximation to the continuous density function I center dot(x), representing mesh width variation. The parameter epsilon (N) = 1/N controls accuracy via the choice of N. For any given grid, a solver provides an error estimate. Taking this as its input, the feedback control law then adjusts the grid, and the interaction continues until the error has been equidistributed. Digital filters may be employed to process the error estimate as well as the density to ensure the regularity of the grid. Once I center dot(x) is determined, another control law determines N based on the prescribed tolerance TOL. The paper focuses on the interaction between control system and solver, and the controller's ability to produce a nearoptimal grid in a stable manner as well as correctly predict how many grid points are needed. Numerical tests demonstrate the advantages of the new control system within the bvpsuite solver, ceteris paribus, for a selection of problems and over a wide range of tolerances. The control system is modular and can be adapted to other solvers and error criteria. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/2536498
 author
 Pulverer, G; Söderlind, Gustaf ^{LU} and Weinmüller, E
 organization
 publishing date
 2011
 type
 Contribution to journal
 publication status
 published
 subject
 keywords
 Grid generation, Error equidistribution, Boundary value problems, Adaptivity, Grid refinement, Step size control, Singular problems, Ordinary differential equations, Singularly perturbed problems
 in
 Numerical Algorithms
 volume
 56
 issue
 1
 pages
 61  92
 publisher
 Springer
 external identifiers

 wos:000285155700005
 scopus:78650222221
 ISSN
 15729265
 DOI
 10.1007/s1107501093740
 language
 English
 LU publication?
 yes
 id
 76fb11e9b8df41bb8b55d1ecec7b806c (old id 2536498)
 date added to LUP
 20120508 11:58:12
 date last changed
 20190106 04:57:47
@article{76fb11e9b8df41bb8b55d1ecec7b806c, abstract = {Grid adaptation in twopoint boundary value problems is usually based on mapping a uniform auxiliary grid to the desired nonuniform grid. Here we combine this approach with a new control system for constructing a grid density function I center dot(x). The local mesh width Delta x (j + 1/2) = x (j + 1) aEuro parts per thousand x (j) with 0 = x (0) < x (1) < ... < x (N) = 1 is computed as Delta x (j + 1/2) = epsilon (N) / phi (j + 1/2), where {phi j+1/2}(0) (N1) is a discrete approximation to the continuous density function I center dot(x), representing mesh width variation. The parameter epsilon (N) = 1/N controls accuracy via the choice of N. For any given grid, a solver provides an error estimate. Taking this as its input, the feedback control law then adjusts the grid, and the interaction continues until the error has been equidistributed. Digital filters may be employed to process the error estimate as well as the density to ensure the regularity of the grid. Once I center dot(x) is determined, another control law determines N based on the prescribed tolerance TOL. The paper focuses on the interaction between control system and solver, and the controller's ability to produce a nearoptimal grid in a stable manner as well as correctly predict how many grid points are needed. Numerical tests demonstrate the advantages of the new control system within the bvpsuite solver, ceteris paribus, for a selection of problems and over a wide range of tolerances. The control system is modular and can be adapted to other solvers and error criteria.}, author = {Pulverer, G and Söderlind, Gustaf and Weinmüller, E}, issn = {15729265}, keyword = {Grid generation,Error equidistribution,Boundary value problems,Adaptivity,Grid refinement,Step size control,Singular problems,Ordinary differential equations,Singularly perturbed problems}, language = {eng}, number = {1}, pages = {6192}, publisher = {Springer}, series = {Numerical Algorithms}, title = {Automatic Grid Control in Adaptive BVP Solvers}, url = {http://dx.doi.org/10.1007/s1107501093740}, volume = {56}, year = {2011}, }