Appendix: Linear Solvers
Linear solvers are used in the stream code to solve the linearized
equations at each iteration in the solution process at a time step.
Because the governing equations in both codes are solved in a sequential
fashion (as opposed to a coupled fashion in most density-based methods),
one can specify a different linear solver for each of the governing
equations in the system of Navier-Stokes equations (momentum, pressure,
energy, etc.) The five linear solvers which are currently available are
shown in the table below. To select a linear solver for a given equation, one
must use the linearSolver option in the corresponding equation options
run control file variable. In order to facilitate the selection of
linear solver parameters for all governing equations, three global run
control file variables can be specified as follows (with default values
shown):
linearSolverTolerance: 1.0e-02
petscSolverName: GMRES
hypreSolverName: AMG
These global values are used as defaults for all governing equations, but they can be overridden. Below is an example of overriding the global default for the momentum equation.
momentumEquationOptions: <linearSolver=SGS, relaxationFactor=0.5, maxIterations=5,
linearSolverTolerance=1.0e-03, petscSolverName=CG>
Local equation override is not currently supported for the
hypreSolverName variable. As shown in the table below, only the LSGS, PETSC and
HYPRE linear solvers use the specified linear solver tolerance to assess
convergence. For these three solvers, the parameter maxIterations
(specified for each governing equation) indicates the maximum number of
iterations that will be performed before the solver kicks out. In
general, the specified linear solver tolerance may be met before the
maximum number of iterations is taken, in which case the solver
terminates, declaring convergence. By contrast, the SGS and FSGS linear
solvers will always perform the maximum number of iterations specified
because the linear solver tolerance is not used.
Solver |
Description |
Solver Names Supported |
Solver Tolerance Used |
|---|---|---|---|
SGS |
Gauss-Seidel |
N/A |
No |
FSGS |
Cache-Optimized Gauss-Seidel |
N/A |
No |
LSGS |
Line Gauss-Seidel |
N/A |
Yes |
PETSC |
PETSc Library |
GMRES, CG, BICG, CGS, BCGS |
Yes |
HYPRE |
HYPRE Library |
GMRES, AM |
Yes |
When using the HYPRE solver, the variable hypreStrongThreshold is used
as follows (default value shown):
hypreStrongThreshold: 0.5
to set the value of the neighbor strong threshold, which has an important effect on the generation of the coarse multi-grid levels. One should generally use the highest value possible for this variable (range 0.25- 0.95) while still maintaining solution stability to substantially reduce CPU time. A starting value of 0.9 is recommended.