Multiplicative cascades applied to PDEs (two numerical examples)

Numerical approximations to the Fourier transformed solution of partial differential equations are obtained via Monte Carlo simulation of certain random multiplicative cascades. Two particular equations are considered: linear diffusion equation and viscous Burgers equation. The algorithms proposed e...

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Bibliographic Details
Published in:Journal of computational physics 2006-05, Vol.214 (1), p.122-136
Main Author: Ramirez, Jorge M.
Format: Article
Language:English
Subjects:
Approximation
Burgers equation
Cascades
Computational techniques
Computer simulation
Exact sciences and technology
Fourier analysis
Fourier space
Linear diffusion
Mathematical analysis
Mathematical methods in physics
Mathematical models
Monte Carlo method
Monte Carlo methods
Partial differential equations
Physics
Random multiplicative cascades
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Summary:Numerical approximations to the Fourier transformed solution of partial differential equations are obtained via Monte Carlo simulation of certain random multiplicative cascades. Two particular equations are considered: linear diffusion equation and viscous Burgers equation. The algorithms proposed exploit the structure of the branching random walks in which the multiplicative cascades are defined. The results show initial numerical approximations with errors less than 5% in the leading Fourier coefficients of the solution. This approximation is then improved substantially using a Picard iteration scheme on the integral equation associated with the representation of the respective PDE in Fourier space.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2005.09.006