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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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
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Abstract	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.
AbstractList	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. 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 Δt 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.
Author	Chen, Dangxing Huang, Jingfang Brandon, Namdi Qu, Wenzhen Kress, Tyler
Author_xml	– sequence: 1 givenname: Wenzhen surname: Qu fullname: Qu, Wenzhen organization: Department of Engineering Mechanics, College of Mechanics and Materials, Hohai University – sequence: 2 givenname: Namdi surname: Brandon fullname: Brandon, Namdi organization: Department of Mathematics, University of North Carolina – sequence: 3 givenname: Dangxing surname: Chen fullname: Chen, Dangxing organization: Department of Mathematics, University of North Carolina – sequence: 4 givenname: Jingfang surname: Huang fullname: Huang, Jingfang email: huang@email.unc.edu organization: Department of Mathematics, University of North Carolina – sequence: 5 givenname: Tyler surname: Kress fullname: Kress, Tyler organization: Department of Mathematics, University of North Carolina
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References_xml	– reference: BrownPHindmarshAPetzoldLUsing Krylov methods in the solution of large-scale differential-algebraic systemsSIAM J. Sci. Comput.19941514671488129862510.1137/09150880812.65060 – reference: AuzingerWHofstatterHKreuzerWWeinmullerEModified defect correction algorithms for ODEs. Part I: general theoryNumer. Algorithms200436135156206287010.1023/B:NUMA.0000033129.73715.7f1058.65068 – reference: HairerEWannerGSolving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems1996BerlinSpringer10.1007/978-3-642-05221-70859.65067 – reference: Hairer, E., Hairer, M.: Gnicodes—matlab programs for geometric numerical integration. In: Blowey, J., Craig, A., Shardlow, T. (eds.) Frontiers in Numerical Analysis, pp. 199–240. Springer, Berlin (2003) – reference: HuangJJiaJMinionMAccelerating the convergence of spectral deferred correction methodsJ. Comput. 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SubjectTerms	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
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Title	A Numerical Framework for Integrating Deferred Correction Methods to Solve High Order Collocation Formulations of ODEs
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