Monte Carlo Finite Volume Element Methods for the Convection-Diffusion Equation with a Random Diffusion Coefficient

Qian Zhang,Zhiyue Zhang

doi:10.1155/2014/642470

Abstract

The paper presents a framework for the construction of Monte Carlo finite volume element method (MCFVEM) for the convection-diffusion equation with a random diffusion coefficient, which is described as a random field. We first approximate the continuous stochastic field by a finite number of random variables via the Karhunen-Loève expansion and transform the initial stochastic problem into a deterministic one with a parameter in high dimensions. Then we generate independent identically distributed approximations of the solution by sampling the coefficient of the equation and employing finite volume element variational formulation. Finally the Monte Carlo (MC) method is used to compute corresponding sample averages. Statistic error is estimated analytically and experimentally. A quasi-Monte Carlo (QMC) technique with Sobol sequences is also used to accelerate convergence, and experiments indicate that it can improve the efficiency of the Monte Carlo method.

Highlights

Mathematical models and computer simulations are widely used in engineering and science
We focus our study on the convectiondiffusion equation with homogeneous Dirichlet boundary conditions
Our goal is to compute the quantity of interest (QoI): Q (u (x, T, ω)) = ∫ u (x, T, ω) dx

Summary

Introduction

Mathematical models and computer simulations are widely used in engineering and science. When the size of noise is relatively small, a perturbation method may be the most popular nonstatistical method, where the random field is expanded via Taylor series around its mean and truncated at certain order. This approach has been used extensively in various engineering fields [1,2,3,4]. Another approach is the Neumann expansion, which is based on the inverse of the stochastic operator in a Neumann series [5, 6]. Their applicability is often strongly dependent on the underlying operator and is typically limited to static problems

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Methods

Conclusion