CMSFlow Matrix Solver: Difference between revisions
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The selection of an iterative solver is one key issue concerning the overall per-formance of the model. CMS has three iteration solvers available, and they are described in more detail below: 1) GMRES variation, 2) BiCGStab, and 3) Gauss-Seidel. The default iterative solver for CMS is a variation of the GMRES (Generalized Minimum Residual) method (Saad 1993) and is used to solve the algebraic equations. | The selection of an iterative solver is one key issue concerning the overall per-formance of the model. CMS has three iteration solvers available, and they are described in more detail below: 1) GMRES variation, 2) BiCGStab, and 3) Gauss-Seidel. The default iterative solver for CMS is a variation of the GMRES (Generalized Minimum Residual) method (Saad 1993) and is used to solve the algebraic equations. | ||
[[File:CMSFlow Model Parameters GeneralTab Matrix Solver V13.2.12.png|500px|thumb|right|CMS Flow General Tab Matrix Solvers ]] | [[File:CMSFlow Model Parameters GeneralTab Matrix Solver V13.2.12.png|500px|thumb|right|CMS Flow General Tab Matrix Solvers ]] | ||
=Gauss-Seiden= | |||
The simplest iterative solver implemented in CMS is the point-implict Gauss-Seidel solver. This method may be applied in CMS with or without Succesive-Over-Relaxation to speed-up convergence (Patankar 1980). Even though the Gauss-Seidel method requires more iterations for convergence, the overall efficiency may be higher than the GMRES and BiCGStab because each itera-tion is computationally inexpensive and the code is parallelized. However, the GMRES and BiCGStab methods are more robust and perform better for large time-steps. | The simplest iterative solver implemented in CMS is the point-implict Gauss-Seidel solver. This method may be applied in CMS with or without Succesive-Over-Relaxation to speed-up convergence (Patankar 1980). Even though the Gauss-Seidel method requires more iterations for convergence, the overall efficiency may be higher than the GMRES and BiCGStab because each itera-tion is computationally inexpensive and the code is parallelized. However, the GMRES and BiCGStab methods are more robust and perform better for large time-steps. | ||
=Gauss-Seiden SOR= | |||
=BiCStab= | |||
The BiCGStab (BiConjugate Gradient Stabilized) iterative solver is also a Krylov subspace solver and is applicable to symmetric and non-symmetrix matrices (Saad 1996). BiCGStab also uses ILUT as a preconditioner (Saad 1994). The BiCGStab method can be viewed as a combination of the standard Biconjugate Gradient solver where each iterative step is followed by a restarted GMRES iterative step. One advantage of the BiCGStab iterative solver is that the memory requirements are constant for each iteration and there are less computational costs when compared to the GMRES method (for less than four iterations). | The BiCGStab (BiConjugate Gradient Stabilized) iterative solver is also a Krylov subspace solver and is applicable to symmetric and non-symmetrix matrices (Saad 1996). BiCGStab also uses ILUT as a preconditioner (Saad 1994). The BiCGStab method can be viewed as a combination of the standard Biconjugate Gradient solver where each iterative step is followed by a restarted GMRES iterative step. One advantage of the BiCGStab iterative solver is that the memory requirements are constant for each iteration and there are less computational costs when compared to the GMRES method (for less than four iterations). | ||
=GMRES= | |||
The original GMRES method (Saad and Schultz 1986) utilizes the Arnoldi process to reduce the coefficient matrix to the Hessenburg form and minimizes the norm of the residual vector over a Krylov subspace at each iterative step. The variation of the GMRES method recommended by Saad (1993) allows changes in preconditioning at every iteration step. An Incomplete LU Factorization (ILUT; Saad, 1994) is used as the preconditioner to speed-up convergence. The GMRES solver is applicable to symmetric and non-symmetric matrices and leads to the smallest residual for a fixed number of iterations. However, the memory requirements and computational costs become increasingly expensive for larger systems. | The original GMRES method (Saad and Schultz 1986) utilizes the Arnoldi process to reduce the coefficient matrix to the Hessenburg form and minimizes the norm of the residual vector over a Krylov subspace at each iterative step. The variation of the GMRES method recommended by Saad (1993) allows changes in preconditioning at every iteration step. An Incomplete LU Factorization (ILUT; Saad, 1994) is used as the preconditioner to speed-up convergence. The GMRES solver is applicable to symmetric and non-symmetric matrices and leads to the smallest residual for a fixed number of iterations. However, the memory requirements and computational costs become increasingly expensive for larger systems. | ||
Revision as of 19:16, 12 December 2022
The selection of an iterative solver is one key issue concerning the overall per-formance of the model. CMS has three iteration solvers available, and they are described in more detail below: 1) GMRES variation, 2) BiCGStab, and 3) Gauss-Seidel. The default iterative solver for CMS is a variation of the GMRES (Generalized Minimum Residual) method (Saad 1993) and is used to solve the algebraic equations.
Gauss-Seiden
The simplest iterative solver implemented in CMS is the point-implict Gauss-Seidel solver. This method may be applied in CMS with or without Succesive-Over-Relaxation to speed-up convergence (Patankar 1980). Even though the Gauss-Seidel method requires more iterations for convergence, the overall efficiency may be higher than the GMRES and BiCGStab because each itera-tion is computationally inexpensive and the code is parallelized. However, the GMRES and BiCGStab methods are more robust and perform better for large time-steps.
Gauss-Seiden SOR
BiCStab
The BiCGStab (BiConjugate Gradient Stabilized) iterative solver is also a Krylov subspace solver and is applicable to symmetric and non-symmetrix matrices (Saad 1996). BiCGStab also uses ILUT as a preconditioner (Saad 1994). The BiCGStab method can be viewed as a combination of the standard Biconjugate Gradient solver where each iterative step is followed by a restarted GMRES iterative step. One advantage of the BiCGStab iterative solver is that the memory requirements are constant for each iteration and there are less computational costs when compared to the GMRES method (for less than four iterations).
GMRES
The original GMRES method (Saad and Schultz 1986) utilizes the Arnoldi process to reduce the coefficient matrix to the Hessenburg form and minimizes the norm of the residual vector over a Krylov subspace at each iterative step. The variation of the GMRES method recommended by Saad (1993) allows changes in preconditioning at every iteration step. An Incomplete LU Factorization (ILUT; Saad, 1994) is used as the preconditioner to speed-up convergence. The GMRES solver is applicable to symmetric and non-symmetric matrices and leads to the smallest residual for a fixed number of iterations. However, the memory requirements and computational costs become increasingly expensive for larger systems.
Reference: CMS Flow Numerical Models