Numerical Simulations of Conservation Laws in Random Heterogeneous Media

Abstract

Conservation laws play an important role in many areas of natural science. When we design numerical schemes for conservation laws, we usually assume that initial data and flux function (the rate of change of quantity of interest) are known exactly. However,this is generally not the case as these are often obtained through indirect measurements.As a consequence the initial data and flux function are known only in terms of statistical quantities like mean, variance and involve some uncertainty. These uncertain inputs should be handled statistically. In our study, we analyze and implement the Monte Carlo Finite Volume Method and the Stochastic Finite Volume Method to solve conservation laws inrandom media. We particularly focus on Stochastic Finite Volume Method and formulate an algorithm for conservation laws with random initial data. Our simulations include that of the inviscid Burgers equation with random inputs.