# Lund University Publications

## LUND UNIVERSITY LIBRARIES

### Automatic Grid Control in Adaptive BVP Solvers

(2011) In Numerical Algorithms 56(1). p.61-92
Abstract
Grid adaptation in two-point 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) (N-1) 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 two-point 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) (N-1) 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 near-optimal 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:
author
organization
publishing date
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
1572-9265
DOI
10.1007/s11075-010-9374-0
language
English
LU publication?
yes
id
76fb11e9-b8df-41bb-8b55-d1ecec7b806c (old id 2536498)
date added to LUP
2012-05-08 11:58:12
date last changed
2019-03-05 01:29:57
```@article{76fb11e9-b8df-41bb-8b55-d1ecec7b806c,
abstract     = {Grid adaptation in two-point 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) &lt; x (1) &lt; ... &lt; x (N) = 1 is computed as Delta x (j + 1/2) = epsilon (N) / phi (j + 1/2), where {phi j+1/2}(0) (N-1) 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 near-optimal 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         = {1572-9265},
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        = {61--92},
publisher    = {Springer},
series       = {Numerical Algorithms},
title        = {Automatic Grid Control in Adaptive BVP Solvers},
url          = {http://dx.doi.org/10.1007/s11075-010-9374-0},
volume       = {56},
year         = {2011},
}

```