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Approximation of stochastic advection diffusion equations with stochastic alternating direction explicit methods. (English) Zbl 1289.65007
Stochastic diffusion and stochastic advection diffusion equations are studied from the numerical point of view. Finite difference methods are presented both for deterministic and stochastic equations. The driving process in stochastic equations is the real-valued time-dependent Wiener process. Stability, consistency and convergence of several alternating direction explicit schemes is proved.
Alternating direction explicit finite difference methods make use of two approximations that are implemented for computations proceeding in alternating directions, e.g., from left to right and from right to left, with each approximation being explicit in its respective direction of computation. Namely, Saul’yev’s and Liu’s numerical schemes are considered for diffusion equations and the Saul’yev/Robert and Weiss scheme is used for advection diffusion equations. Convergence and stability of these methods is known for deterministic PDEs from [V. K. Saul’ev, Dokl. Akad. Nauk SSSR 115, 1077–1079 (1957; Zbl 0080.11104); S.-L. Liu, “Stable explicit difference approximations to parabolic partial differential equations”, AIChE J. 15, No. 3, 334–338 (1969); K. V. Roberts and N. O. Weiss, Math. Comput. 20, 272–299 (1966; Zbl 0137.33404)]. Their adoption to stochastic partial differential equations (Section 3) is relatively new and continuously follows from earlier work [A. R. Soheili et al., Bull. Iran. Math. Soc. 37, No. 2, Part 1, 61–83 (2011; Zbl 1260.60124)].
An a priory estimate (Section 4) is used to proof the stability of all studied schemes in terms of Fourier analysis (Section 5), the consistency conditions are checked and the mean square consistency is proofed using Itō’s isometry (Section 6), and the convergence of the schemes is proved using the Taylor expansion (Section 7). Computational efficiency is then presented on two particular examples with known expectation of the solution (Section 8).
The authors apply standard tools from stochastic and numerical analysis, which may be interesting from the methodological point of view; the results themselves may be interesting for specialists working in the field of stochastic partial differential equations as well as for applied mathematicians dealing with models based on diffusion and advection diffusion equations.

MSC:
65C30 Numerical solutions to stochastic differential and integral equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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