Convergence

In a recent article [2] Dutt, Greengard and Rohklin define two new methods of deferred correction. Convergence for the the first method, using one step methods, has been proven in Hansen [8]. In this paper we augment the theory presented in [8] and use this to prove convergence for the second deferred correction method in [2] using linear k-step methods. This method has been known as Spectral Deferred Correction.

[1]  Christoph W. Ueberhuber,et al.  Iterated defect correction for the efficient solution of stiff systems of ordinary differential equations , 1977 .

[2]  M. Minion Semi-implicit spectral deferred correction methods for ordinary differential equations , 2003 .

[3]  Klaus Böhmer,et al.  Defect Correction Methods , 1984, Computing Supplementum.

[4]  Robert D. Skeel,et al.  A Theoretical Framework for Proving Accuracy Results for Deferred Corrections , 1982 .

[5]  E. Hairer On the order of iterated defect correction , 1978 .

[6]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[7]  E. Hairer,et al.  Solving Ordinary Differential Equations I , 1987 .

[8]  Christoph W. Ueberhuber,et al.  An extension of the applicability of iterated deffered corrections , 1977 .

[9]  Christoph W. Ueberhuber,et al.  Iterated defect correction for differential equations part I: theoretical results , 2005, Computing.

[10]  H. Stetter Analysis of Discretization Methods for Ordinary Differential Equations , 1973 .

[11]  L. Greengard,et al.  Spectral Deferred Correction Methods for Ordinary Differential Equations , 2000 .

[12]  Bengt Lindberg Error estimation and iterative improvement for discretization algorithms , 1980 .

[13]  F. Krogh,et al.  Solving Ordinary Differential Equations , 2019, Programming for Computations - Python.

[14]  P. Zadunaisky On the estimation of errors propagated in the numerical integration of ordinary differential equations , 1976 .

[15]  Michael L. Minion,et al.  Conservative multi-implicit spectral deferred correction methods for reacting gas dynamics , 2004 .

[16]  A. Bourlioux,et al.  High-order multi-implicit spectral deferred correction methods for problems of reactive flow , 2003 .