### Abstract

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 stiff 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.

Original language | English (US) |
---|---|

Pages (from-to) | 241-266 |

Number of pages | 26 |

Journal | BIT Numerical Mathematics |

Volume | 40 |

Issue number | 2 |

State | Published - Jun 2000 |

### Fingerprint

### Keywords

- Deferred correction
- Initial value problems
- Spectral methods
- Stiffness

### ASJC Scopus subject areas

- Computer Graphics and Computer-Aided Design
- Software
- Applied Mathematics
- Computational Mathematics

### Cite this

*BIT Numerical Mathematics*,

*40*(2), 241-266.

**Spectral deferred correction methods for ordinary differential equations.** / Dutt, Alok; Greengard, Leslie; Rokhlin, Vladimir.

Research output: Contribution to journal › Article

*BIT Numerical Mathematics*, vol. 40, no. 2, pp. 241-266.

}

TY - JOUR

T1 - Spectral deferred correction methods for ordinary differential equations

AU - Dutt, Alok

AU - Greengard, Leslie

AU - Rokhlin, Vladimir

PY - 2000/6

Y1 - 2000/6

N2 - 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 stiff 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.

AB - 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 stiff 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.

KW - Deferred correction

KW - Initial value problems

KW - Spectral methods

KW - Stiffness

UR - http://www.scopus.com/inward/record.url?scp=0042185232&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0042185232&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0042185232

VL - 40

SP - 241

EP - 266

JO - BIT Numerical Mathematics

JF - BIT Numerical Mathematics

SN - 0006-3835

IS - 2

ER -