Numerical Treatment of the Stochastic Advection-Diffusion Equation Using the Spectral Stochastic Techniques

Maha Hamed • I. L. El-Kalla • Beih S. El-Desouky • Mohamed A. El-Beltagy


Stochastic advection diffusion equation (SADE) with multiplicative stochastic input is a practical mathematical model for different physical phenomena. In this paper, SADE will be studied using two spectral stochastic techniques. The first is the Wiener chaos expansion (WCE) technique and the second is the Wiener-Hermite expansion with perturbation (WHEP) technique. These techniques convert the SADE into a system of deterministic partial differential equation (DPDE) that can be solved using a deterministic numerical method which is suitable for the periodic boundary conditions. Convergence analysis is discussed and some of the second order moments are compared. The numerical results demonstrate the efficiency of both techniques. The WCE technique is more accuracy than the WHEP technique. The diffusion and advection coefficient and the intensity of Gaussian white noise play important roles in the SADE solution. The study shows that the WCE technique is more practical to get the closed form mean solution while the WHEP technique gets the mean solution in the form of an infinite series.




International Journal of Engineering Research and Science

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