Approximation of stochastic advection diffusion equations with Stochastic Alternating Direction Explicit methods

• Published in: Applications of mathematics (Prague) Vol. 58; no. 4; pp. 439 - 471
• Main Authors: Soheili, Ali R., Arezoomandan, Mahdieh
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2013; Springer Nature B.V
• Subjects: Accuracy; Algorithms; Analysis; Applications of Mathematics; Applied mathematics; Approximation; Classical and Continuum Physics; Diffusion; Finite difference method; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical models; Mathematics; Mathematics and Statistics; Methods; Noise; Numerical analysis; Optimization; Partial differential equations; Stochasticity; Studies; Theorems; Theoretical; stochastic partial differential equation; convergence; 65M06; alternating direction method; Saul’yev method; Liu method; finite difference method; consistency; 60H15; stability
• ISSN: 0862-7940; 1572-9109
• DOI: 10.1007/s10492-013-0022-6

The numerical solutions of stochastic partial differential equations of Itô type with time white noise process, using stable stochastic explicit finite difference methods are considered in the paper. Basically, Stochastic Alternating Direction Explicit (SADE) finite difference schemes for solving stochastic time dependent advection-diffusion and diffusion equations are represented and the main properties of these stochastic numerical methods, e.g. stability, consistency and convergence are analyzed. In particular, it is proved that when stable alternating direction explicit schemes for solving linear parabolic PDEs are developed to the stochastic case, they retain their unconditional stability properties applying to stochastic advection-diffusion and diffusion SPDEs. Numerically, unconditional stable SADE techniques are significant for approximating the solutions of the proposed SPDEs because they do not impose any restrictions for refining the computational domains. The performance of the proposed methods is tested for stochastic diffusion and advection-diffusion problems, and the accuracy and efficiency of the numerical methods are demonstrated.