research-article

Algorithm 962: BACOLI: B-spline Adaptive Collocation Software for PDEs with Interpolation-Based Spatial Error Control

Authors:

Paul MuirAuthors Info & Claims

ACM Transactions on Mathematical Software (TOMS), Volume 42, Issue 3

Article No.: 25, Pages 1 - 17

https://doi.org/10.1145/2818312

Published: 23 May 2016 Publication History

Abstract

BACOL and BACOLR are (Fortran 77) B-spline adaptive collocation packages for the numerical solution of 1D parabolic Partial Differential Equations (PDEs). The packages have been shown to be superior to other similar packages, especially for problems exhibiting sharp, moving spatial layer regions, where a stringent tolerance is imposed. In addition to providing temporal error control through the timestepping software, BACOL and BACOLR feature control of a high-order estimate of the spatial error of the approximate solution, obtained by computing a second approximate solution of one higher order of accuracy; the cost is substantial—execution time and memory usage are almost doubled.

In this article, we discuss BACOLI, a new version of BACOL that computes only one approximate solution and uses efficient interpolation-based schemes to obtain a spatial error estimate. In previous studies these schemes have been shown to provide spatial error estimates of comparable quality to those of BACOL. We describe the substantial modification of BACOL needed to obtain BACOLI, and provide numerical results showing that BACOLI is significantly more efficient than BACOL, in some cases by as much as a factor of 2. We also introduce a Fortran 95 wrapper for BACOLI (called BACOLI95) and discuss its simplified user interface.

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Cited By

Adams MMuir P(2024)Differential equation software for the computation of error-controlled continuous approximate solutionsNumerical Algorithms10.1007/s11075-024-01784-196:3(1021-1044)Online publication date: 1-Jul-2024
https://dl.acm.org/doi/10.1007/s11075-024-01784-1
Green KSpiteri R(2021)Solving Cardiac Bidomain Problems with B-spline Adaptive CollocationRecent Developments in Mathematical, Statistical and Computational Sciences10.1007/978-3-030-63591-6_28(301-311)Online publication date: 30-Aug-2021
https://doi.org/10.1007/978-3-030-63591-6_28
Tannahill CMuir P(2021)bacoli_py—A Python Package for the Error Controlled Numerical Solution of 1D Time-Dependent PDEsRecent Developments in Mathematical, Statistical and Computational Sciences10.1007/978-3-030-63591-6_27(289-300)Online publication date: 30-Aug-2021
https://doi.org/10.1007/978-3-030-63591-6_27
Show More Cited By

Index Terms

Algorithm 962: BACOLI: B-spline Adaptive Collocation Software for PDEs with Interpolation-Based Spatial Error Control
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
      1. Partial differential equations

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In this paper we describe BACOL, a high-order, spatially and temporally adaptive software package for solving systems of one-dimensional parabolic partial differential equations and then compare it with several related software packages. BACOL employs ...

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Published In

cover image ACM Transactions on Mathematical Software

ACM Transactions on Mathematical Software Volume 42, Issue 3

June 2016

208 pages

ISSN:0098-3500

EISSN:1557-7295

DOI:10.1145/2935754

Editor:
Michael A. Heroux
Sandia National Laboratories, USA

Issue’s Table of Contents

Copyright © 2016 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

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Association for Computing Machinery

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Publication History

Published: 23 May 2016

Accepted: 01 August 2015

Revised: 01 March 2015

Received: 01 February 2014

Published in TOMS Volume 42, Issue 3

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Mathematics of Information Technology and Complex Systems Network
Saint Mary's University
Natural Sciences and Engineering Research Council of Canada

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Cited By

Adams MMuir P(2024)Differential equation software for the computation of error-controlled continuous approximate solutionsNumerical Algorithms10.1007/s11075-024-01784-196:3(1021-1044)Online publication date: 1-Jul-2024
https://dl.acm.org/doi/10.1007/s11075-024-01784-1
Green KSpiteri R(2021)Solving Cardiac Bidomain Problems with B-spline Adaptive CollocationRecent Developments in Mathematical, Statistical and Computational Sciences10.1007/978-3-030-63591-6_28(301-311)Online publication date: 30-Aug-2021
https://doi.org/10.1007/978-3-030-63591-6_28
Tannahill CMuir P(2021)bacoli_py—A Python Package for the Error Controlled Numerical Solution of 1D Time-Dependent PDEsRecent Developments in Mathematical, Statistical and Computational Sciences10.1007/978-3-030-63591-6_27(289-300)Online publication date: 30-Aug-2021
https://doi.org/10.1007/978-3-030-63591-6_27
Green KSpiteri R(2019)Extended BACOLIACM Transactions on Mathematical Software10.1145/330132045:1(1-19)Online publication date: 14-Mar-2019
https://dl.acm.org/doi/10.1145/3301320
Adams MTannahill CMuir P(2019)Error control Gaussian collocation software for boundary value ODEs and 1D time-dependent PDEsNumerical Algorithms10.1007/s11075-019-00738-281:4(1505-1519)Online publication date: 1-Aug-2019
https://dl.acm.org/doi/10.1007/s11075-019-00738-2
Pew JLi ZTannahill CMuir PFairweather G(2018)Performance analysis of error-control B-spline Gaussian collocation software for PDEsComputers & Mathematics with Applications10.1016/j.camwa.2018.11.025Online publication date: Dec-2018
https://doi.org/10.1016/j.camwa.2018.11.025

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