### Abstract

This paper deals with the numerical solution of boundary value problems of ordinary differential equations posed on infinite intervals. We cut the infinite interval at a finite, large enough point and insert additional, so-called asymptotic boundary conditions at the far (right) end and then solve the resulting two-point boundary value problem by an-stable symmetric collocation method. Problems arise, because standard theory predicts the use of many grid points as the length of the interval increases. Using the exponential decay of the 'infinite' solution, an 'asymptotic' a priori mesh-size sequence which increases exponentially, and which therefore only employs a reasonable number of meshpoints, is developed and stability, as the length of the interval tends to infinity, is shown. We also show that the condition number of the collocation equations is asymptotically proportional to the number of meshpoints employed when using this exponentially graded mesh. Using £-stage collocation at Gaussian points and requiring an accuracy O() at the knots implies that the number of meshpoints is 0(^{-1/2k}) as 0.

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

Pages (from-to) | 123-150 |

Number of pages | 28 |

Journal | Mathematics of Computation |

Volume | 40 |

Issue number | 161 |

DOIs | |

State | Published - 1983 |

Externally published | Yes |

### Fingerprint

### Keywords

- Asymptotic properties
- Difference equations
- Nonlinear boundary value problems
- Singular points
- Stability of difference equations

### ASJC Scopus subject areas

- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics

### Cite this

*Mathematics of Computation*,

*40*(161), 123-150. https://doi.org/10.1090/S0025-5718-1983-0679437-X

**Collocation methods for boundary value problems on long intervals.** / Markowich, Peter A.; Ringhofer, Christian.

Research output: Contribution to journal › Article

*Mathematics of Computation*, vol. 40, no. 161, pp. 123-150. https://doi.org/10.1090/S0025-5718-1983-0679437-X

}

TY - JOUR

T1 - Collocation methods for boundary value problems on long intervals

AU - Markowich, Peter A.

AU - Ringhofer, Christian

PY - 1983

Y1 - 1983

N2 - This paper deals with the numerical solution of boundary value problems of ordinary differential equations posed on infinite intervals. We cut the infinite interval at a finite, large enough point and insert additional, so-called asymptotic boundary conditions at the far (right) end and then solve the resulting two-point boundary value problem by an-stable symmetric collocation method. Problems arise, because standard theory predicts the use of many grid points as the length of the interval increases. Using the exponential decay of the 'infinite' solution, an 'asymptotic' a priori mesh-size sequence which increases exponentially, and which therefore only employs a reasonable number of meshpoints, is developed and stability, as the length of the interval tends to infinity, is shown. We also show that the condition number of the collocation equations is asymptotically proportional to the number of meshpoints employed when using this exponentially graded mesh. Using £-stage collocation at Gaussian points and requiring an accuracy O() at the knots implies that the number of meshpoints is 0(-1/2k) as 0.

AB - This paper deals with the numerical solution of boundary value problems of ordinary differential equations posed on infinite intervals. We cut the infinite interval at a finite, large enough point and insert additional, so-called asymptotic boundary conditions at the far (right) end and then solve the resulting two-point boundary value problem by an-stable symmetric collocation method. Problems arise, because standard theory predicts the use of many grid points as the length of the interval increases. Using the exponential decay of the 'infinite' solution, an 'asymptotic' a priori mesh-size sequence which increases exponentially, and which therefore only employs a reasonable number of meshpoints, is developed and stability, as the length of the interval tends to infinity, is shown. We also show that the condition number of the collocation equations is asymptotically proportional to the number of meshpoints employed when using this exponentially graded mesh. Using £-stage collocation at Gaussian points and requiring an accuracy O() at the knots implies that the number of meshpoints is 0(-1/2k) as 0.

KW - Asymptotic properties

KW - Difference equations

KW - Nonlinear boundary value problems

KW - Singular points

KW - Stability of difference equations

UR - http://www.scopus.com/inward/record.url?scp=84966259415&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84966259415&partnerID=8YFLogxK

U2 - 10.1090/S0025-5718-1983-0679437-X

DO - 10.1090/S0025-5718-1983-0679437-X

M3 - Article

AN - SCOPUS:84966259415

VL - 40

SP - 123

EP - 150

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 161

ER -