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Abstract <p>Solvers of systems of linear algebraic equations based on domain decomposition and multigrid methods demonstrate high efficiency in solving two-dimensional problems using cell-centered methods with a 5-point discretization stencil. In the 3D case, such methods utilize a 7-point stencil, requiring modification of the solvers. The aim of this work is to develop similar solvers for systems arising from the solution of three-dimensional problems using cell-centered methods with a 7-point discretization stencil. Formulas for computing the elements of the system matrices in three-dimensional subdomains and on interface planes, considering the 7-point stencil, are derived. It is shown that the structure of the interface system matrix coincides with the two-dimensional case. Therefore, the same direct solver used for 2D can be applied. Solving systems within subdomains is performed using the FGMRES method with preconditioning. When using the multigrid method as a preconditioner, certain parts of the algorithm require modifications for the transition from 2D to 3D. An algorithm for 3D ADLJ-smoother is presented. Since the considered system is derived from cell-centered grid methods where unknowns are located at cell centers rather than grid nodes, non-standard projection and prolongation operators are required for transferring between grid levels. The general principle for their constructing is demonstrated for 1D. Based on these formulas, three-dimensional operators are constructed for all possible cases. The developed solvers were verified on systems formed when solving three-dimensional hydrodynamics problems using the LS-STAG-3D immersed boundary cell-centered method with a 7-point stencil. For comparison, similar solvers for the two-dimensional case were also considered, applied to systems of the same dimension but formed from solving two-dimensional hydrodynamics problems using the LS-STAG method. Computational experiments show that the required accuracy is achieved in a comparable number of iterations for 2D and 3D problems of the same system dimension.</p>

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Development of Solvers for Systems of Linear Algebraic Equations for Cell-Centered Methods with a 7-Point Discretization Stencil

  • 期刊论文

  • 发表日期：

    2026年5月23日

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  • V. V. Puzikova

摘要

Abstract

Solvers of systems of linear algebraic equations based on domain decomposition and multigrid methods demonstrate high efficiency in solving two-dimensional problems using cell-centered methods with a 5-point discretization stencil. In the 3D case, such methods utilize a 7-point stencil, requiring modification of the solvers. The aim of this work is to develop similar solvers for systems arising from the solution of three-dimensional problems using cell-centered methods with a 7-point discretization stencil. Formulas for computing the elements of the system matrices in three-dimensional subdomains and on interface planes, considering the 7-point stencil, are derived. It is shown that the structure of the interface system matrix coincides with the two-dimensional case. Therefore, the same direct solver used for 2D can be applied. Solving systems within subdomains is performed using the FGMRES method with preconditioning. When using the multigrid method as a preconditioner, certain parts of the algorithm require modifications for the transition from 2D to 3D. An algorithm for 3D ADLJ-smoother is presented. Since the considered system is derived from cell-centered grid methods where unknowns are located at cell centers rather than grid nodes, non-standard projection and prolongation operators are required for transferring between grid levels. The general principle for their constructing is demonstrated for 1D. Based on these formulas, three-dimensional operators are constructed for all possible cases. The developed solvers were verified on systems formed when solving three-dimensional hydrodynamics problems using the LS-STAG-3D immersed boundary cell-centered method with a 7-point stencil. For comparison, similar solvers for the two-dimensional case were also considered, applied to systems of the same dimension but formed from solving two-dimensional hydrodynamics problems using the LS-STAG method. Computational experiments show that the required accuracy is achieved in a comparable number of iterations for 2D and 3D problems of the same system dimension.