A Study of Some Finite Difference Schemes for a Unidirectional Stochastic Transport Equation

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Abstract
We study a unidirectional hyperbolic transport equation, with a homogeneous stochastic transport velocity, solved by Monte Carlo simulation. Several finite difference schemes are applied to the deterministic problem in each Monte Carlo iteration, and the numerical solution of the stochastic problem is compared to analytical solutions derived in the paper. We present both a theoretical analysis and summarized results from extensive numerical experiments. The behavior of the various schemes depends on the stochastic properties of the problem, and there are new demands on the schemes when they are used as part of a Monte Carlo simulation. For example, schemes that are very oscillatory for a single deterministic problem, like the leapfrog scheme, turn out to be efficient and accurate for the corresponding stochastic problem.