Spectral deferred correction methods for ordinary differential equations

被引:357
作者
Dutt, A
Greengard, L
Rokhlin, V
机构
[1] Bank Amer, London E1 8DE, England
[2] NYU, Courant Inst Math Sci, New York, NY 10012 USA
[3] Yale Univ, Dept Math, New Haven, CT 06520 USA
[4] Yale Univ, Dept Comp Sci, New Haven, CT 06520 USA
来源
BIT | 2000年 / 40卷 / 02期
关键词
spectral methods; initial value problems; deferred correction; stiffness;
D O I
10.1023/A:1022338906936
中图分类号
TP31 [计算机软件];
学科分类号
081202 ; 0835 ;
摘要
We introduce a new class of methods for the Cauchy problem for ordinary differential equations (ODEs). We begin by converting the original ODE into the corresponding Picard equation and apply a deferred correction procedure in the integral formulation, driven by either the explicit or the implicit Euler marching scheme. The approach results in algorithms of essentially arbitrary order accuracy for both non-stiff and stiff problems; their performance is illustrated with several numerical examples. For non-stiff problems, the stability behavior of the obtained explicit schemes is very satisfactory and algorithms with orders between 8 and 20 should be competitive with the best existing ones. In our preliminary experiments with sti problems, a simple adaptive implementation of the method demonstrates performance comparable to that of a state-of-the-art extrapolation code (at least, at moderate to high precision). Deferred correction methods based on the Picard equation appear to be promising candidates for further investigation.
引用
收藏
页码:241 / 266
页数:26
相关论文
共 21 条
[1]  
Bohmer K., 1984, DEFECT CORRECTION ME
[2]  
Brenan KE, 1995, NUMERICAL SOLUTION I
[3]  
Butcher J. C., 1987, The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods
[4]   RECENT PROGRESS IN EXTRAPOLATION METHODS FOR ORDINARY DIFFERENTIAL-EQUATIONS [J].
DEUFLHARD, P .
SIAM REVIEW, 1985, 27 (04) :505-535
[5]   Fast algorithms for polynomial interpolation, integration, and differentiation [J].
Dutt, A ;
Gu, M ;
Rokhlin, V .
SIAM JOURNAL ON NUMERICAL ANALYSIS, 1996, 33 (05) :1689-1711
[6]  
Frank R., 1977, BIT (Nordisk Tidskrift for Informationsbehandling), V17, P146, DOI 10.1007/BF01932286
[7]  
Gear C. W., 1971, NUMERICAL INITIAL VA
[8]  
Gottlieb D., 1977, NUMERICAL ANAL SPECT
[9]   SPECTRAL INTEGRATION AND 2-POINT BOUNDARY-VALUE-PROBLEMS [J].
GREENGARD, L .
SIAM JOURNAL ON NUMERICAL ANALYSIS, 1991, 28 (04) :1071-1080
[10]  
Hairer E., 2008, Solving Ordinary Differential Equations I Nonstiff problems