SOLVERS

The Solvers SimScene contains set of tests for all the linear-equation solvers which have been introduced into Earth.

The tests concern steady heat conduction through a medium with temperature-independent conductivity and no convection. Such tasks can, in principle be colved within a single sweep. Very often however more sweeps are required as a consequence of round-off errors

. Each task can be solved using various linear equation solvers at the same conditions. Tasks use as structured as unstructured grid.

After running a task user is able to see parameters of solver as dependencies maximum residual, maximum corrections, sum of residuals, sum of corrections and internal parameters of solver vs iteration number.

User can see the dependence temperature near the cold and hot objects and in the middle of domain from sweep to sweep. Besides user is able to analyze convergence due to nett-sources of objects and sum of them, which mean the balance of heat fluxes for these tasks. Last method is more correct for physical purposes.

The full list of solvers is:

- default, which is Stone-like solver with block-correction, but without print-out;
- Conjugate Residual Solver (KIVA-II) with several pre-conditioners;
- MADI, which is modification of well-known ADI solver;
- PHOENICS's solver (PHSOL), which is the same as default, but with detailed print-out;
- Conjugate Gradient (SK_CG) is standard conjugate gradient solver with some pre-conditioners (from SPARSEKIT);
- Conjugate Gradient Method (Normal Residual equation) (CGNR) with some pre-conditioners (from SPARSEKIT);
- Bi-Conjugate Gradient Method (BCG) with some pre-conditioners (from SPARSEKIT);
- Bi-Conjugate Gradient Method with partial pivoting (DBCG) with some pre-conditioners (from SPARSEKIT);
- Bi-Conjugate Gradient Method stabilized (BCGSTAB) with some pre-conditioners (from SPARSEKIT);
- Transpose-Free Quasi-Minimum Residual method (TFQMR) with some pre-conditioners (from SPARSEKIT);
- Full Orthogonalization Method (FOM) with some pre-conditioners (from SPARSEKIT);
- Generalized Minimum RESidual method (GMRES) with some pre-conditioners (from SPARSEKIT);
- Flexible version of Generalized Minimum RESidual method (FGMRES) with some pre-conditioners (from SPARSEKIT);
- Direct versions of Quasi Generalize Minimum Residual method (DQGMRES) with some pre-conditioners (from SPARSEKIT);
- Preconditioned GMRES solver (PGMRES) with ILUT pre-conditioner (from SPARSEKIT);
- Conjugate Residual solver (CNGR) with Jacobi or ILU pre-conditioners;
- Modified Strongly Implicit Procedure (MSIP).

The most of solvers can utilise pre-conditioners. Solvers from package SPARSEKIT can use following pre-conditioners each:

- Incomplete LU factorization with dual truncation strategy (ILUT);
- ILU with single dropping + diagonal compensation (~MILUT) (ILUD);
- level-k ILU (ILUK);
- simple ILU(0) preconditioning (ILU(0)), which is recommended by author for testing purposes only;
- MILU(0) preconditioning (MILU(0)), which is recommended by author for testing purposes only.

Conjugate Residual Solver (KIVA-II) can be used with all pre-conditioners from SPARSEKIT, with solvers PHSOL and MADI as pre-conditioners and with point-by-point and Jackobi solvers as pre-conditioners.

The main purpose of creation SimScene SOLVERS is preparing the tool for testing of linear equation solvers. Well know fact is that rate of convergence depends on quality of linear equation solver. From one hand non-linear equations of hydrodynamics tasks do not require exact solution of linear system on each sweep, because linear eqautions has not exact coefficient, but from other hand, better performance of linear equation solver leads to fast convergence even for non-linear tasks.

The first task for testing of solvers is case "2D conductivity with body in xz plane". User is free for choice to use some body into domian or remain domain empty.

This is conductivity task with constant conductivity, where cold temperature is set on the west side and hot temperature is set on east side. This task has not exact solution if some body is placed in the centre of domain. Convergence is not always good, if conductivity of body is strongly different from conductivity of domain material (air). Therefore, it is good test for convergence of linear equation solvers.

As a example the view of domain with cylinder as body is demonstrated on the following figure:Implementations of this case for structured grid and ustructured grid have some differencies. Therefore, descriptions of menu items are different. See Menu items (SP) and Menu items (USP) accordingly.