Approximation of stochastic advection diffusion equations with stochastic alternating direction explicit methods.

*(English)*Zbl 1289.65007Stochastic 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.

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.

Reviewer: Jan Pospíšil (Plzeň)

##### 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 |

##### Keywords:

stochastic partial differential equation; finite difference method; alternating direction method; Saul’ev method; Liu method
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\textit{A. R. Soheili} and \textit{M. Arezoomandan}, Appl. Math., Praha 58, No. 4, 439--471 (2013; Zbl 1289.65007)

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##### References:

[1] | Allen, E. J.; Novosel, S. J.; Zhang, Z., Finite element and difference approximation of some linear stochastic partial differential equations, Stochastics Stochastics Rep., 64, 117-142, (1998) · Zbl 0907.65147 |

[2] | W.F. Ames: Numerical Methods for Partial Differential Equations. 3. ed. Computer Science and Scientific Computing. Academic Press, Boston, 1992. |

[3] | Campbell, L. J.; Yin, B., On the stability of alternating-direction explicit methods for advection-diffusion equations, Numer. Methods Partial Differ. Equations, 23, 1429-1444, (2007) · Zbl 1129.65058 |

[4] | Davie, A.M.; Gaines, J.G., Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations, Math. Comput., 70, 121-134, (2001) · Zbl 0956.60064 |

[5] | Higham, D. J., An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43, 525-546, (2001) · Zbl 0979.65007 |

[6] | P.E. Kloeden, E. Platen: Numerical Solution of Stochastic Differential Equations. Applications of Mathematics 23. Springer, Berlin, 1992. · Zbl 0752.60043 |

[7] | Komori, Y.; Mitsui, T., Stable ROW-type weak scheme for stochastic differential equations, Monte Carlo Methods Appl., 1, 279-300, (1995) · Zbl 0841.65146 |

[8] | Liu, S. L., Stable explicit difference approximations to parabolic partial differential equations, AIChE J., 15, 334-338, (1969) |

[9] | S. McDonald: Finite difference approximation for linear stochastic partial differential equation with method of lines. MPRA Paper No. 3983 (2006). http://mpra.ub.uni-muenchen.de/3983. · Zbl 0907.65147 |

[10] | G.N. Milstein: Numerical Integration of Stochastic Differential Equations. Transl. from the Russian. Mathematics and its Applications 313. Kluwer Academic Publishers, Dordrecht, 1994. · Zbl 0810.65144 |

[11] | Rößler, A., Stochastic Taylor expansions for the expectation of functionals of diffusion processes, Stochastic Anal. Appl., 22, 1553-1576, (2004) · Zbl 1065.60068 |

[12] | Rößler, A.; Seaïd, M.; Zahri, M., Method of lines for stochastic boundary-value problems with additive noise, Appl. Math. Comput., 199, 301-314, (2008) · Zbl 1142.65007 |

[13] | Roth, C., Difference methods for stochastic partial differential equations, Z. Angew. Math. Mech., 82, 821-830, (2002) · Zbl 1010.60057 |

[14] | C. Roth: Approximations of Solution of a First Order Stochastic Partial Differential Equation, Report. Institut Optimierung und Stochastik, Universität Halle-Wittenberg, Halle, 1989. |

[15] | Saul’yev, V.K.; Stewart, K.L. (ed.), Integration of equations of parabolic type by the method of nets, (1964), Oxford · Zbl 0128.11803 |

[16] | Saul’yev, V.K., On a method of numerical integration of a diffusion equation, Dokl. Akad. Nauk SSSR, 115, 1077-1080, (1957) |

[17] | Soheili, A.R.; Niasar, M.B.; Arezoomandan, M., Approximation of stochastic parabolic differential equations with two different finite difference schemes, Bull. Iran. Math. Soc., 37, 61-83, (2011) · Zbl 1260.60124 |

[18] | J.C. Strikwerda: Finite difference schemes and partial differential equations. 2nd ed. Society for Industrial and Applied Mathematics, Philadelphia, 2004. · Zbl 1071.65118 |

[19] | J.W. Thomas: Numerical Partial Differential Equations: Finite Difference Methods. Texts in Applied Mathematics 22. Springer, New York, 1995. · Zbl 0831.65087 |

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