The Neumann–Ulam Scheme and Dominant-Part Extraction of the Operator for the Integral Equation of the Two-Phase Filtration Problem: Theory, Unbiased Estimators, and a Computational Experiment

This study addresses the theoretical development and computational implementation of probabilistic-statistical methods for solving the stationary problems of two-phase filtration of two immiscible, incompressible fluids in a porous medium. The object of investigation is a second-kind Fredholm integral equation, denoted Equation, which arises from applying the Levi function to the regular filtration problem. It is proved that, provided ax≡a=const>0, the norm of the integral operator in the space CΩ‾ is strictly less than unity, which guarantees the existence, uniqueness, and convergence of successive approximations. Building on this result, a Neumann–Ulam scheme is constructed: the generating Markov chain is described, a probabilistic representation of the solution is derived, and the asymptotic unbiasedness of the idealized estimators and the bounded finite εδ-bias and finite variance of the practically implementable estimators are established. A generalization of the Neumann–Ulam scheme is proposed and theoretically justified—a dominant-part extraction method for the operator—which affords a substantial reduction in variance whenever the norm of the residual integral operator is small. A computational experiment includes a variable-coefficient test case to examine the practical behavior of the proposed methods beyond the constant-coefficient setting considered in the theoretical results. The proposed algorithms are shown to outperform the standard finite-difference scheme in the accuracy of the solution at an individual point of the domain, an advantage that becomes particularly pronounced for multidimensional problems.