Local Discontinuous Galerkin Method for the Stochastic Heat Equation

In this paper we study the Local Discontinuous Galerkin scheme for solving the stochastic heat equation driven by the space white noise. We begin by giving a brief introduction to stochastic processes, stochastic differential equations, and their importance in the modern mathematical context. From there, using an example stochastic elliptic partial differential equation, we approximate the white noise term using piecewise constant functions and show that it will also hold for the stochastic heat equation. We give an introduction to Local Discontinuous Galerkin method and produce a block matrix equation by separating the stochastic heat equation into two first order partial differential equations. We prove that the stochastic heat equation has a unique solution since its expected value converges to the heat equation without the white noise term. From there, we give a possible numerical way of solving the matrix equation as well as how to handle the stochastic term in this numerical method. After solving the matrix equation, we discuss what the average is of all the equations that result from the matrix equation. CONTENTS