### Abstract

A new Monte Carlo algorithm for constructing and sampling stationary isotropic Gaussian random fields with power-law energy spectrum, infrared divergence, and fractal self-similar scaling is developed here. The theoretical basis for this algorithm involves the fact that such a random field is well approximated by a superposition of random one-dimensional plane waves involving a fixed finite number of directions. In general each one-dimensional plane wave is the sum of a random shear layer and a random acoustical wave. These one-dimensional random plane waves are then simulated by a wavelet Monte Carlo method for a single space variable developed recently by the authors. The computational results reported in this paper demonstrate remarkable low variance and economical representation of such Gaussian random fields through this new algorithm. In particular, the velocity structure function for an imcorepressible isotropic Gaussian random field in two space dimensions with the Kolmogoroff spectrum can be simulated accurately over 12 decades with only 100 realizations of the algorithm with the scaling exponent accurate to 1.1% and the constant prefactor accurate to 6%; in fact, the exponent of the velocity structure function can be computed over 12 decades within 3.3% with only 10 realizations. Furthermore, only 46,592 active computational elements are utilized in each realization to achieve these results for 12 decades of scaling behavior.

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

Pages (from-to) | 146-162 |

Number of pages | 17 |

Journal | Journal of Computational Physics |

Volume | 117 |

Issue number | 1 |

DOIs | |

State | Published - 1995 |

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### ASJC Scopus subject areas

- Computer Science Applications
- Physics and Astronomy(all)
- Physics and Astronomy (miscellaneous)

### Cite this

**A new algorithm with plane waves and wavelets for random velocity fields with many spatial scales.** / Elliott, Frank W.; Majda, Andrew J.

Research output: Contribution to journal › Article

*Journal of Computational Physics*, vol. 117, no. 1, pp. 146-162. https://doi.org/10.1006/jcph.1995.1052

}

TY - JOUR

T1 - A new algorithm with plane waves and wavelets for random velocity fields with many spatial scales

AU - Elliott, Frank W.

AU - Majda, Andrew J.

PY - 1995

Y1 - 1995

N2 - A new Monte Carlo algorithm for constructing and sampling stationary isotropic Gaussian random fields with power-law energy spectrum, infrared divergence, and fractal self-similar scaling is developed here. The theoretical basis for this algorithm involves the fact that such a random field is well approximated by a superposition of random one-dimensional plane waves involving a fixed finite number of directions. In general each one-dimensional plane wave is the sum of a random shear layer and a random acoustical wave. These one-dimensional random plane waves are then simulated by a wavelet Monte Carlo method for a single space variable developed recently by the authors. The computational results reported in this paper demonstrate remarkable low variance and economical representation of such Gaussian random fields through this new algorithm. In particular, the velocity structure function for an imcorepressible isotropic Gaussian random field in two space dimensions with the Kolmogoroff spectrum can be simulated accurately over 12 decades with only 100 realizations of the algorithm with the scaling exponent accurate to 1.1% and the constant prefactor accurate to 6%; in fact, the exponent of the velocity structure function can be computed over 12 decades within 3.3% with only 10 realizations. Furthermore, only 46,592 active computational elements are utilized in each realization to achieve these results for 12 decades of scaling behavior.

AB - A new Monte Carlo algorithm for constructing and sampling stationary isotropic Gaussian random fields with power-law energy spectrum, infrared divergence, and fractal self-similar scaling is developed here. The theoretical basis for this algorithm involves the fact that such a random field is well approximated by a superposition of random one-dimensional plane waves involving a fixed finite number of directions. In general each one-dimensional plane wave is the sum of a random shear layer and a random acoustical wave. These one-dimensional random plane waves are then simulated by a wavelet Monte Carlo method for a single space variable developed recently by the authors. The computational results reported in this paper demonstrate remarkable low variance and economical representation of such Gaussian random fields through this new algorithm. In particular, the velocity structure function for an imcorepressible isotropic Gaussian random field in two space dimensions with the Kolmogoroff spectrum can be simulated accurately over 12 decades with only 100 realizations of the algorithm with the scaling exponent accurate to 1.1% and the constant prefactor accurate to 6%; in fact, the exponent of the velocity structure function can be computed over 12 decades within 3.3% with only 10 realizations. Furthermore, only 46,592 active computational elements are utilized in each realization to achieve these results for 12 decades of scaling behavior.

UR - http://www.scopus.com/inward/record.url?scp=0010811852&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0010811852&partnerID=8YFLogxK

U2 - 10.1006/jcph.1995.1052

DO - 10.1006/jcph.1995.1052

M3 - Article

AN - SCOPUS:0010811852

VL - 117

SP - 146

EP - 162

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 1

ER -