### Abstract

In this paper we consider both analytically and numerically several finite-dimensional approximations for the inviscid Burgers-Hopf equation. Fourier Galerkin truncation is introduced and studied as a simple one-dimensional model with intrinsic chaos and a well-defined mathematical structure allowing for an equilibrium statistical mechanics formalism. A simple scaling theory for correlations is developed that is supported strongly by the numerical evidence. Several semi-discrete difference schemes with similar mathematical properties conserving discrete momentum and energy are also considered. The mathematical properties of the difference schemes are analyzed and the behavior of the difference schemes is compared and contrasted with the Fourier Galerkin truncation. Numerical simulations are presented which show similarities and subtle differences between different finite-dimensional approximations both in the deterministic and stochastic regimes with many degrees of freedom.

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

Pages (from-to) | 39-96 |

Number of pages | 58 |

Journal | Milan Journal of Mathematics |

Volume | 70 |

Issue number | 1 |

State | Published - 2002 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Milan Journal of Mathematics*,

*70*(1), 39-96.

**Statistical mechanics for truncations of the Burgers-Hopf equation : A model for intrinsic stochastic behavior with scaling.** / Majda, A.; Timofeyev, I.

Research output: Contribution to journal › Article

*Milan Journal of Mathematics*, vol. 70, no. 1, pp. 39-96.

}

TY - JOUR

T1 - Statistical mechanics for truncations of the Burgers-Hopf equation

T2 - A model for intrinsic stochastic behavior with scaling

AU - Majda, A.

AU - Timofeyev, I.

PY - 2002

Y1 - 2002

N2 - In this paper we consider both analytically and numerically several finite-dimensional approximations for the inviscid Burgers-Hopf equation. Fourier Galerkin truncation is introduced and studied as a simple one-dimensional model with intrinsic chaos and a well-defined mathematical structure allowing for an equilibrium statistical mechanics formalism. A simple scaling theory for correlations is developed that is supported strongly by the numerical evidence. Several semi-discrete difference schemes with similar mathematical properties conserving discrete momentum and energy are also considered. The mathematical properties of the difference schemes are analyzed and the behavior of the difference schemes is compared and contrasted with the Fourier Galerkin truncation. Numerical simulations are presented which show similarities and subtle differences between different finite-dimensional approximations both in the deterministic and stochastic regimes with many degrees of freedom.

AB - In this paper we consider both analytically and numerically several finite-dimensional approximations for the inviscid Burgers-Hopf equation. Fourier Galerkin truncation is introduced and studied as a simple one-dimensional model with intrinsic chaos and a well-defined mathematical structure allowing for an equilibrium statistical mechanics formalism. A simple scaling theory for correlations is developed that is supported strongly by the numerical evidence. Several semi-discrete difference schemes with similar mathematical properties conserving discrete momentum and energy are also considered. The mathematical properties of the difference schemes are analyzed and the behavior of the difference schemes is compared and contrasted with the Fourier Galerkin truncation. Numerical simulations are presented which show similarities and subtle differences between different finite-dimensional approximations both in the deterministic and stochastic regimes with many degrees of freedom.

UR - http://www.scopus.com/inward/record.url?scp=0002023117&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0002023117&partnerID=8YFLogxK

M3 - Article

VL - 70

SP - 39

EP - 96

JO - Milan Journal of Mathematics

JF - Milan Journal of Mathematics

SN - 1424-9286

IS - 1

ER -