Statistical moments of the random linear transport equation

This paper deals with a numerical scheme to approximate the mth moment of the solution of the one-dimensional random linear transport equation. The initial condition is assumed to be a random function and the transport velocity is a random variable. The scheme is based on local Riemann problem solutions and Godunov's method. We show that the scheme is stable and consistent with an advective-diffusive equation. Numerical examples are added to illustrate our approach.

[1]  Fábio Antonio Dorini,et al.  Statistical moments of the solution of the random Burgers-Riemann problem , 2009, Math. Comput. Simul..

[2]  G. Schuëller.  A state-of-the-art report on computational stochastic mechanics , 1997 .

[3]  Fábio Antonio Dorini,et al.  A numerical scheme for the variance of the solution of the random transport equation , 2007, Appl. Math. Comput..

[4]  Maurice G. Kendall.  Conditions for Uniqueness in the Problem of Moments , 1940 .

[5]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

[6]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[7]  Fabio Antonio Dorini,et al.  A note on the Riemann problem for the random transport equation , 2007 .

[8]  Myron B. Allen,et al.  Groundwater Transport With StochasticRetardation: Numerical Results , 1998 .

[9]  René Carmona,et al.  Stochastic Partial Differential Equations: Six Perspectives , 1998 .

[10]  I. Bohachevsky,et al.  Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics , 1959 .

[11]  Ronald L. Wasserstein,et al.  Monte Carlo: Concepts, Algorithms, and Applications , 1997 .

[12]  Bernt Øksendal,et al.  Stochastic differential equations (3rd ed.): an introduction with applications , 1992 .

[13]  You‐Kuan Zhang.  Stochastic Methods for Flow in Porous Media: Coping with Uncertainties , 2001 .

[14]  J. Strikwerda.  Finite Difference Schemes and Partial Differential Equations , 1989 .

[15]  N. Bershad,et al.  Random differential equations in science and engineering , 1975, Proceedings of the IEEE.

[16]  B. Øksendal.  Stochastic differential equations : an introduction with applications , 1987 .

[17]  Fábio Antonio Dorini,et al.  A finite volume method for the mean of the solution of the random transport equation , 2007, Appl. Math. Comput..

[18]  James Glimm,et al.  Stochastic partial differential equations: Selected applications in continuum physics , 1999 .

[19]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[20]  A. Polyanin.  Handbook of Linear Partial Differential Equations for Engineers and Scientists , 2001 .