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Extending the study on the linear advection equation subject to stochastic velocity field and initial condition

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  • Calatayud, J.
  • Cortés, J.-C.
  • Dorini, F.A.
  • Jornet, M.

Abstract

In this paper we extend the study on the linear advection equation with independent stochastic velocity and initial condition performed in Dorini and Cunha (2011). By using both existing and novel results on the stochastic chain rule, we solve the random linear advection equation in the mean square sense. We provide a new expression for the probability density function of the solution stochastic process, which can be computed as accurately as wanted via Monte Carlo simulations, and which does not require the specific probability distribution of the integral of the velocity. This allows us to solve the non-Gaussian velocity case, which was not treated in the aforementioned contribution. Several numerical results illustrate the computations of the probability density function by using our approach. On the other hand, we derive a theoretical partial differential equation for the probability density function of the solution stochastic process. Finally, a shorter and easier derivation of the joint probability density function of the response process at two spatial points is obtained by applying conditional expectations appropriately.

Suggested Citation

  • Calatayud, J. & Cortés, J.-C. & Dorini, F.A. & Jornet, M., 2020. "Extending the study on the linear advection equation subject to stochastic velocity field and initial condition," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 172(C), pages 159-174.
    • Handle: RePEc:eee:matcom:v:172:y:2020:i:c:p:159-174
    DOI: 10.1016/j.matcom.2019.12.014
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    References listed on IDEAS

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    1. El-Wakil, S.A. & Sallah, M. & El-Hanbaly, A.M., 2015. "Random variable transformation for generalized stochastic radiative transfer in finite participating slab media," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 435(C), pages 66-79.
    2. Dorini, F.A. & Cunha, M.C.C., 2011. "On the linear advection equation subject to random velocity fields," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(4), pages 679-690.
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