Abstract
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.
| Original language | English |
|---|---|
| Pages (from-to) | 122-136 |
| Number of pages | 15 |
| Journal | Journal of Computational Physics |
| Volume | 214 |
| Issue number | 1 |
| DOIs | |
| State | Published - May 1 2006 |
| Externally published | Yes |
Funding
This work was initiated in the author’s MS thesis under the direction of Enrique Thomann. The author also expresses gratitude to the faculty in the Focussed Research Group at Oregon State University for advice and encouragement. Special thanks to Ed. Waymire for his helpful comments. This work was partially supported by Focussed Research Group collaborative awards DMS-0073958 and DMS-0073865 to Oregon State University by the National Science Foundation, and by CMG collaborative awards DMS-0327705 to Oregon State University.
Keywords
- Burgers equation
- Fourier space
- Linear diffusion
- Monte Carlo method
- Random multiplicative cascades