A finite volume method for scalar conservation laws with stochastic time-space dependent flux functions

Authors:
Kamel Mohamed;Mohammed Seaid;Mostafa Zahri
Affiliations:
Department of Computer Science, Faculty of Applied Sciences, University of Taibah Madinah, Saudi Arabia;School of Engineering and Computing Sciences, University of Durham, South Road DH1 3LE, UK;Department of Mathematics, Faculty of Sciences, University of Taibah, P.O. Box 344 Madinah, Saudi Arabia
Venue:
Journal of Computational and Applied Mathematics
Year:
2013

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Abstract

We propose a new finite volume method for scalar conservation laws with stochastic time-space dependent flux functions. The stochastic effects appear in the flux function and can be interpreted as a random manner to localize the discontinuity in the time-space dependent flux function. The location of the interface between the fluxes can be obtained by solving a system of stochastic differential equations for the velocity fluctuation and displacement variable. In this paper we develop a modified Rusanov method for the reconstruction of numerical fluxes in the finite volume discretization. To solve the system of stochastic differential equations for the interface we apply a second-order Runge-Kutta scheme. Numerical results are presented for stochastic problems in traffic flow and two-phase flow applications. It is found that the proposed finite volume method offers a robust and accurate approach for solving scalar conservation laws with stochastic time-space dependent flux functions.