### Abstract

Original language | English |
---|---|

Pages (from-to) | 964-976 |

Number of pages | 12 |

Journal | SIAM Journal on Scientific Computing |

Volume | 10 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1989 |

### Fingerprint

### Keywords

- Adams PECE method
- interpolant
- defect
- numerical mathematics

### Cite this

*SIAM Journal on Scientific Computing*,

*10*(5), 964-976. https://doi.org/10.1137/0910056

}

*SIAM Journal on Scientific Computing*, vol. 10, no. 5, pp. 964-976. https://doi.org/10.1137/0910056

**Defect estimation in Adams PECE codes.** / Higham, D.J.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Defect estimation in Adams PECE codes

AU - Higham, D.J.

PY - 1989

Y1 - 1989

N2 - Many modern codes for solving the nonstiff initial value problem $y'(x) - f(x,y(x)) = 0,y(a)$ given, $a leqq x leqq b$, produce, in addition to a discretised solution, a function $p(x)$ that approximates $y(x)$ over $[a,b]$. The associated defect $delta (x): = p'(x) - f(x,p(x))$ is a natural measure of the error. In this paper the problem of reliably estimating the defect in Adams PECE methods is considered. Attention is focused on the widely used Shampine-Gordon variable order, variable step code fitted with a continuously differentiable interpolant $p(x)$ due to Watts and Shampine [SIAM .J. Sci. Statist. Comput, 7 (1986), pp. 334-345]. It is shown that over each step an asymptotically correct estimate of the defect can be obtained by sampling at a single, suitably chosen point. It is also shown that a valid "free" estimate can be formed without recourse to sampling. Numerical results are given to support the theory.

AB - Many modern codes for solving the nonstiff initial value problem $y'(x) - f(x,y(x)) = 0,y(a)$ given, $a leqq x leqq b$, produce, in addition to a discretised solution, a function $p(x)$ that approximates $y(x)$ over $[a,b]$. The associated defect $delta (x): = p'(x) - f(x,p(x))$ is a natural measure of the error. In this paper the problem of reliably estimating the defect in Adams PECE methods is considered. Attention is focused on the widely used Shampine-Gordon variable order, variable step code fitted with a continuously differentiable interpolant $p(x)$ due to Watts and Shampine [SIAM .J. Sci. Statist. Comput, 7 (1986), pp. 334-345]. It is shown that over each step an asymptotically correct estimate of the defect can be obtained by sampling at a single, suitably chosen point. It is also shown that a valid "free" estimate can be formed without recourse to sampling. Numerical results are given to support the theory.

KW - Adams PECE method

KW - interpolant

KW - defect

KW - numerical mathematics

U2 - 10.1137/0910056

DO - 10.1137/0910056

M3 - Article

VL - 10

SP - 964

EP - 976

JO - SIAM Journal on Scientific Computing

JF - SIAM Journal on Scientific Computing

SN - 1064-8275

IS - 5

ER -