### Abstract

We present a numerical scheme for the approximation of nonlinear evolution equations over large time intervals. Our algorithm is motivated from the dynamical systems point of view. In particular, we adapt the methodology of approximate inertial manifolds to a finite difference scheme. This leads to a differential treatment in which the higher (i.e. unresolved) modes are expressed in terms of the lower modes. As a particular example we derive an approximate inertial manifold for Burgers' equation and develop a numerical algorithm suitable for computing. We perform a parameter study in which we compare the accuracy of a standard scheme with our modified scheme. For all values of the parameters (which are the coefficient of viscosity and the cell size), we obtain a decrease in the numerical error by at least a factor of 2.0 with the modified scheme. The decrease in error is substantially greater over large regions of the parameters space.

Original language | English (US) |
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Pages (from-to) | 175-184 |

Number of pages | 10 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 60 |

Issue number | 1-4 |

DOIs | |

State | Published - Nov 1 1992 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Applied Mathematics
- Statistical and Nonlinear Physics

### Cite this

*Physica D: Nonlinear Phenomena*,

*60*(1-4), 175-184. https://doi.org/10.1016/0167-2789(92)90234-E

**An approximate inertial manifold for computing Burgers' equation.** / Margolin, L. G.; Jones, Donald.

Research output: Contribution to journal › Article

*Physica D: Nonlinear Phenomena*, vol. 60, no. 1-4, pp. 175-184. https://doi.org/10.1016/0167-2789(92)90234-E

}

TY - JOUR

T1 - An approximate inertial manifold for computing Burgers' equation

AU - Margolin, L. G.

AU - Jones, Donald

PY - 1992/11/1

Y1 - 1992/11/1

N2 - We present a numerical scheme for the approximation of nonlinear evolution equations over large time intervals. Our algorithm is motivated from the dynamical systems point of view. In particular, we adapt the methodology of approximate inertial manifolds to a finite difference scheme. This leads to a differential treatment in which the higher (i.e. unresolved) modes are expressed in terms of the lower modes. As a particular example we derive an approximate inertial manifold for Burgers' equation and develop a numerical algorithm suitable for computing. We perform a parameter study in which we compare the accuracy of a standard scheme with our modified scheme. For all values of the parameters (which are the coefficient of viscosity and the cell size), we obtain a decrease in the numerical error by at least a factor of 2.0 with the modified scheme. The decrease in error is substantially greater over large regions of the parameters space.

AB - We present a numerical scheme for the approximation of nonlinear evolution equations over large time intervals. Our algorithm is motivated from the dynamical systems point of view. In particular, we adapt the methodology of approximate inertial manifolds to a finite difference scheme. This leads to a differential treatment in which the higher (i.e. unresolved) modes are expressed in terms of the lower modes. As a particular example we derive an approximate inertial manifold for Burgers' equation and develop a numerical algorithm suitable for computing. We perform a parameter study in which we compare the accuracy of a standard scheme with our modified scheme. For all values of the parameters (which are the coefficient of viscosity and the cell size), we obtain a decrease in the numerical error by at least a factor of 2.0 with the modified scheme. The decrease in error is substantially greater over large regions of the parameters space.

UR - http://www.scopus.com/inward/record.url?scp=0000055155&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000055155&partnerID=8YFLogxK

U2 - 10.1016/0167-2789(92)90234-E

DO - 10.1016/0167-2789(92)90234-E

M3 - Article

AN - SCOPUS:0000055155

VL - 60

SP - 175

EP - 184

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 1-4

ER -