A Proposed Stochastic Finite Difference Approach Based on Homogenous Chaos Expansion

O H Galal

doi:10.1155/2013/950469

Abstract

This paper proposes a stochastic finite difference approach, based on homogenous chaos expansion (SFDHC). The said approach can handle time dependent nonlinear as well as linear systems with deterministic or stochastic initial and boundary conditions. In this approach, included stochastic parameters are modeled as second-order stochastic processes and are expanded using Karhunen-Loève expansion, while the response function is approximated using homogenous chaos expansion. Galerkin projection is used in converting the original stochastic partial differential equation (PDE) into a set of coupled deterministic partial differential equations and then solved using finite difference method. Two well-known equations were used for efficiency validation of the method proposed. First one being the linear diffusion equation with stochastic parameter and the second is the nonlinear Burger's equation with stochastic parameter and stochastic initial and boundary conditions. In both of these examples, the probability distribution function of the response manifested close conformity to the results obtained from Monte Carlo simulation with optimized computational cost.

Highlights

This paper proposes a stochastic finite difference approach, based on homogenous chaos expansion (SFDHC)
As evidenced in this work, the SFDHC technique was designed to solve a wide range of time dependent nonlinear as well as linear problems with stochastic or deterministic initial and boundary conditions and exposed to deterministic or stochastic excitation
The computation times spent for SFDHC, on a Core-i3 computer, were 3.08 and 3.91 seconds for linear and nonlinear problems, respectively, as against 40.19 and 989.82 seconds for Monte Carlo simulation (MCS), respectively, which means that 7.66% and 0.395% only of the execution time are in favor of SFDHC

Summary

Introduction

These rare studies include Kaminski [3] which introduced a second-order perturbation, second probabilistic moment analysis in the context of FDM This technique uses the perturbation method to expand the input random fields and was applied to time dependent linear problems with a small range of variability, where the first two statistical moments of the response were calculated. This paper utilizes the homogenous chaos expansion in the context of finite difference method (SFDHC) In this technique, the random inputs are discretized using Karhunen-Loeve (KL) expansion, while the response is represented in terms of homogenous chaos (HC) expansion a Galerkin projection scheme is applied to generate a system of deterministic equations, which can be solved using classical deterministic FDM.

Random Field Discretization

Illustrative Example I

Illustrative Example II

Discussion and Conclusions