Finite difference approximation of the pressure equation for fluid flow in a stochastic medium — a probabilistic approach
- 1 January 1996
- journal article
- research article
- Published by Taylor & Francis in Communications in Partial Differential Equations
- Vol. 21 (9-10) , 1367-1388
- https://doi.org/10.1080/03605309608821231
Abstract
In modelling the pressure p(x,ω) at of an incompressible fluid in a heterogeneous, isotropic medium with a stochastic permeability k(x, ω) ≥ 0, Holden, Lindstr⊘m, Øksendal, Ub⊘e and Zhang [HLØUZ95] studied the following stochastic differential equation where ƒ is a given source and ⋄ denotes the Wick product. They proved existence of an explicit solution in (S)-1. In this paper we define a finite difference scheme to approximate the above equation and prove that this scheme converges in (S)-1. To solve the obtained finite difference equation, we prove that an adapted Jacobi iterative method is convergent.Keywords
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