No abstract

This article considered the numerical simulation of multicomponent multiphase flow in porous media. The resulting system of nonlinear equations linearized by the Newton-Raphson method and solved with the iterative Generalized minimal residual method (GMRES) algorithm. To achieve better convergence, we used the ILU(0) preconditioner to the GMRES algorithm. As a result, we used a completely implicit scheme called the Newton-ILU0-GMRES algorithm to solve the problem of interest. Based on the obtained sequential algorithm, we implemented a parallel algorithm using Message Passing Interface (MPI) technology. Additionally, we made comparisons between the parallel program of the presented algorithm and the parallel program using the ready-made Portable Extensible Toolkit for Scientific Computation (PETSc) library. We developed an MPI parallel algorithm and tested it on the MVS-10P supercomputer of the Interdepartmental Supercomputer Center of the Russian Academy of Sciences.

The problem of generalization of the method is the main question that arises when studying the quality of iterative methods. The efficiency of solving systems using iterative methods directly depends on the assumptions about the system of equations to be solved. Prerequisites are used to provide a more efficient solution. Many types of prerequisites are currently known, for example, prerequisites based on the approximation of the system matrix: ILU, IQR, and ILQ; Prerequisites based on the approximation of the inverse matrix: a polynomial, rarely filled approximation of the inverse matrix (for example, AINV), an approximation in the factorized form of the inverse matrix (for example, FSAI, SPAI, etc.). This article analyzes the CG and CG methods with the preconditioner ILU (0) by the example of solving the two-dimensional Poisson equation. The CG method is usually used to solve any system of linear equations. ILU (0) was selected as a prerequisite for the article. The incomplete LU decomposition (ILU (0)) is an efficient precursor and is easily implemented. This suggests a system that can be solved to speed up the accumulation of CG and other iterative methods, that is, to reduce the number of iterations. The ILU (0) preconditioner is very easy to detect using the LU decomposition. Since the linear matrix was rarely filled, the CSR format was used to store the matrix in memory. ILU (0) + CG, i.e. the algorithm with a precondition, was assembled 5-8 times faster than the CG algorithm. Data on the number of iterations of convergence of the method without a preconditioner and with the ILU(0) preconditioner were obtained and analyzed.

Newton’s method has been widely used in simulation multiphase, multicomponent flow in porous media. In addition, to solve systems of linear equations in such problems, the generalized minimal residual method (GMRES) is often used. This paper analyzed the one-dimensional problem of multicomponent fluid flow in a porous medium and solved the system of the algebraic equation with the Newton-GMRES method. We calculated the linear equations with the GMRES, the GMRES with restarts after every m steps—GMRES (m) and preconditioned with Incomplete Lower-Upper factorization, where the factors L and U have the same sparsity pattern as the original matrix—the ILU(0)-GMRES algorithms, respectively, and compared the computation time and convergence. In the course of the research, the influence of the preconditioner and restarts of the GMRES (m) algorithm on the computation time was revealed; in particular, they were able to speed up the program.