# Linear Solver Settings¶

There are several situations where the options selected in zCFD require the solution of a linear system. These are:

When the Time Marching scheme is set to euler implicit.

When RBFs are used for mesh motion.

When the incompressible solver is selected.

When running on CPUs zCFD uses Petsc linear solver library and when running on NVidia GPUs zCFD uses the AMGX linear solver library. Both libraries offer a wide array of solver and preconditioner choices and altering the settings can have a significant impact on the performance and convergence of the solver. Options can be passed to Petsc via the control dictionary and AMGX options are set using a json file which is read at runtime.

## Petsc¶

Details of the available options for the Petsc KSP linear system solvers used by zCFD are given here. these options can be passed through to Petsc via the zCFD control dictionary using the “linear solver options” key. The default values are given below:

```
"linear solver options": {"flow": { "-ksp_type": "fgmres",
"-ksp_rtol": "1.0e-3",
"-ksp_monitor": "",
"-ksp_converged_reason": "" },
"turbulence": { "-ksp_type": "fgmres",
"-ksp_rtol": "1.0e-5",
"-ksp_monitor": "",
"-ksp_converged_reason": "" },
"rbf": { "-ksp_type": "fgmres",
"-ksp_rtol": "1.0e-5",
"-ksp_monitor": "" }
}
```

Settings are provided for the mean flow, turbulence and RBF linear systems. Most of the settings detailed in the Petsc documentation can be passed through by adding them as key, value pairs to the appropriate dictionary. Where an option doesn’t take a value an empty string needs to be provided.

The Hypre library of algebraic multigrid methods is available via Petsc PCHYPRE and the associated settings can be supplied via the “linear solver options” dictionary. Using Hypre’s boomeramg package as a preconditioner has shown good performance with the incompressible solver:

```
"linear solver options": {"flow": { "-ksp_type": "fgmres",
"-ksp_rtol": "1.0e-3",
"-ksp_monitor": "",
"-ksp_converged_reason": "",
"-pc_type": "hypre",
"-pc_hypre_type": "boomeramg"},
...
```

Further Hypre options are detailed in the Petsc PCHYPRE documentation.

## AMGX¶

The AMGX settings for the mean flow, turbulence and RBF linear systems can be found in ZCFD_HOME/amgx.json, ZCFD_HOME/turbamgx.json and ZCFD_HOME/RBF_amgx.json. The mean flow settings are shown below:

```
"config_version": 2,
"determinism_flag": 1,
"solver": {
"preconditioner": {
"error_scaling": 0,
"print_grid_stats": 0,
"max_uncolored_percentage": 0.05,
"algorithm": "AGGREGATION",
"solver": "AMG",
"smoother": "MULTICOLOR_GS",
"presweeps": 0,
"selector": "SIZE_8",
"coarse_solver": "NOSOLVER",
"max_iters": 1,
"postsweeps": 3,
"min_coarse_rows": 32,
"relaxation_factor": 0.75,
"scope": "amg",
"max_levels": 40,
"matrix_coloring_scheme": "PARALLEL_GREEDY",
"cycle": "V"
},
"use_scalar_norm": 1,
"solver": "FGMRES",
"print_solve_stats": 1,
"obtain_timings": 1,
"max_iters": 10,
"monitor_residual": 1,
"gmres_n_restart": 5,
"convergence": "RELATIVE_INI_CORE",
"scope": "main",
"tolerance": 1e-3,
"norm": "L2"
}
```

In general these settings provide good performance for most cases. If issues are encountered converging the linear system reducing the “tolerance” may help. Otherwise altering the multigrid aggregation selection algorithm “selector”: “SIZE_8” to “SIZE_4” or “SIZE_2” will build smaller aggregates and hence increase the number of coarse levels - improving convergence at the expense of memory use. Increasing the number of “gmres_n_restart” may also improve convergence at the expense of increased memory use.

There are too many AMGX options to detail here but example configurations for different solvers are given here.

When running on NVidia GPUs and using AMGX there is only one option available in the “linear solver options” dictionary:

```
"linear solver options": {"double precision": False}
```

The “double precision” option controls whether the mean flow linear system is solved in single or double precision. Solving in single precision brings a reduction in memory use and a speed up to the solver. However, double precision may be required for more challenging cases.