A Numerical Framework for Integrating Deferred Correction Methods to Solve High Order Collocation Formulations of ODEs

Recent analysis and numerical experiments show that the deferred correction methods are competitive numerical schemes for time dependent differential equations. These methods differ in the mathematical formulations, choices of collocation points, and numerical integration or differentiation strategi...

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Bibliographic Details
Published in:Journal of scientific computing Vol. 68; no. 2; pp. 484 - 520
Main Authors: Qu, Wenzhen, Brandon, Namdi, Chen, Dangxing, Huang, Jingfang, Kress, Tyler
Format: Journal Article
Language:English
Published: New York Springer US 01.08.2016
Springer Nature B.V
Subjects:
Algorithms
Approximation
Boundary value problems
Collocation
Computational Mathematics and Numerical Analysis
Convergence
Differential equations
Error correction
Error reduction
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Numerical analysis
Numerical integration
Numerical methods
Ordinary differential equations
Physical properties
Simulation
Theoretical
Time dependence
Time integration
Deferred correction methods
Krylov subspace methods
65M70
Collocation formulations
Preconditioners
65B05
Jacobian-free Newton–Krylov methods
65M12
ISSN:0885-7474, 1573-7691
Online Access:Get full text
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Summary:Recent analysis and numerical experiments show that the deferred correction methods are competitive numerical schemes for time dependent differential equations. These methods differ in the mathematical formulations, choices of collocation points, and numerical integration or differentiation strategies. Existing analyses of these methods usually follow traditional ODE theory and study each algorithm’s convergence and stability properties as the step size Δ t varies. In this paper, we study the deferred correction methods from a different perspective by separating two different concepts in the algorithm: (1) the properties of the converged solution to the collocation formulation, and (2) the convergence procedure utilizing the deferred correction schemes to iteratively and efficiently reduce the error in the provisional solution. This new viewpoint allows the construction of a numerical framework to integrate existing techniques, by (1) selecting an appropriate collocation discretization based on the physical properties of the solution to balance the time step size and accuracy of the initial approximate solution; and by (2) applying different deferred correction strategies for reducing different components in the error of the provisional solution. This paper discusses properties of different components in the numerical framework, and presents preliminary results on the effective integration of these components for ODE initial value problems. Our results provide useful guidelines for implementing “optimal” time integration schemes for general time dependent differential equations.
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ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-015-0146-9