NDLI: Implications of the Choice of Quadrature Nodes for Picard Integral Deferred Corrections Methods for Ordinary Differential Equations

Content Provider	Springer Nature Link
Author	Layton, Anita T. Minion, Michael L.
Copyright Year	2005
Abstract	This paper concerns a class of deferred correction methods recently developed for initial value ordinary differential equations; such methods are based on a Picard integral form of the correction equation. These methods divide a given timestep [tn,tn+1] into substeps, and use function values computed at these substeps to approximate the Picard integral by means of a numerical quadrature. The main purpose of this paper is to present a detailed analysis of the implications of the location of quadrature nodes on the accuracy and stability of the overall method. Comparisons between Gauss-Legendre, Gauss-Lobatto, Gauss-Radau, and uniformly spaced points are presented. Also, for a given set of quadrature nodes, quadrature rules may be formulated that include or exclude function values computed at the left-hand endpoint tn. Quadrature rules that do not depend on the left-hand endpoint (which are referred to as right-hand quadrature rules) are shown to lead to L(α)-stable implicit methods with α≈π/2. The semi-implicit analog of this property is also discussed. Numerical results suggest that the use of uniform quadrature nodes, as opposed to nodes based on Gaussian quadratures, does not significantly affect the stability or accuracy of these methods for orders less than ten. In contrast, a study of the reduction of order for stiff equations shows that when uniform quadrature nodes are used in conjunction with a right-hand quadrature rule, the form and extent of order-reduction changes considerably. Specifically, a reduction of order to $\mathcal{O}(\epsilon^2)$ is observed for uniform nodes as opposed to $\mathcal{O}(\epsilon\Delta{t})$ for non-uniform nodes, where Δt denotes the time step and ε a stiffness parameter such that ε→0 corresponds to the problem becoming increasingly stiff.
Starting Page	341
Ending Page	373
Page Count	33
File Format	PDF
ISSN	00063835
Journal	BIT Numerical Mathematics
Volume Number	45
Issue Number	2
e-ISSN	15729125
Language	English
Publisher	Springer-Verlag
Publisher Date	2005-01-01
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	semi-implicit methods deferred correction methods order reduction Picard integral Numeric Computing Mathematics Computational Mathematics and Numerical Analysis
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computer Networks and Communications Computational Mathematics Software

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

Implications of the choice of quadrature nodes for Picard integral deferred corrections methods for ordinary differential equations

Implications of the Choice of Quadrature Nodes for Picard Integral Deferred Corrections Methods for Ordinary Differential Equations

Implications of the choice of predictors for semi-implicit picard integral deferred corrections methods (2007).

Implications of the choice of predictors for semi-implicit Picard integral deferred correction methods

Implications of the Choice of Predictors for Semi-Implicit Picard Integral Deferred Correction Methods ∗

Implications of the Choice of Predictors for Semi-Implicit Picard Integral Deferred Correction Methods

Spectral Deferred Correction Methods for Ordinary Differential Equations

Semi-implicit Krylov Deferred Correction Methods for Ordinary Differential Equations (2009)

Semi-implicit krylov deferred correction methods for ordinary differential equations (2009).

Implications of the Choice of Quadrature Nodes for Picard Integral Deferred Corrections Methods for Ordinary Differential Equations

Similar Documents

Implications of the choice of quadrature nodes for Picard integral deferred corrections methods for ordinary differential equations

Implications of the Choice of Quadrature Nodes for Picard Integral Deferred Corrections Methods for Ordinary Differential Equations

Implications of the choice of predictors for semi-implicit picard integral deferred corrections methods (2007).

Implications of the choice of predictors for semi-implicit Picard integral deferred correction methods

Implications of the Choice of Predictors for Semi-Implicit Picard Integral Deferred Correction Methods ∗

Implications of the Choice of Predictors for Semi-Implicit Picard Integral Deferred Correction Methods

Spectral Deferred Correction Methods for Ordinary Differential Equations

Semi-implicit Krylov Deferred Correction Methods for Ordinary Differential Equations (2009)

Semi-implicit krylov deferred correction methods for ordinary differential equations (2009).

Implications of the Choice of Quadrature Nodes for Picard Integral Deferred Corrections Methods for Ordinary Differential Equations