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- Title
Numerical Method for a Filtration Model Involving a Nonlinear Partial Integro-Differential Equation.
- Authors
Baigereyev, Dossan; Omariyeva, Dinara; Temirbekov, Nurlan; Yergaliyev, Yerlan; Boranbek, Kulzhamila
- Abstract
In this paper, we propose an efficient numerical method for solving an initial boundary value problem for a coupled system of equations consisting of a nonlinear parabolic partial integro-differential equation and an elliptic equation with a nonlinear term. This problem has an important applied significance in petroleum engineering and finds application in modeling two-phase nonequilibrium fluid flows in a porous medium with a generalized nonequilibrium law. The construction of the numerical method is based on employing the finite element method in the spatial direction and the finite difference approximation to the time derivative. Newton's method and the second-order approximation formula are applied for the treatment of nonlinear terms. The stability and convergence of the discrete scheme as well as the convergence of the iterative process is rigorously proven. Numerical tests are conducted to confirm the theoretical analysis. The constructed method is applied to study the two-phase nonequilibrium flow of an incompressible fluid in a porous medium. In addition, we present two examples of models allowing for prediction of the behavior of a fluid flow in a porous medium that are reduced to solving the nonlinear integro-differential equations studied in the paper.
- Subjects
INTEGRO-differential equations; NONEQUILIBRIUM flow; NEWTON-Raphson method; BOUNDARY value problems; INITIAL value problems; NONLINEAR equations
- Publication
Mathematics (2227-7390), 2022, Vol 10, Issue 8, pN.PAG
- ISSN
2227-7390
- Publication type
Article
- DOI
10.3390/math10081319