Solution Strategy

All physics solvers share a common solution strategy for nonlinear time-dependent problems. Here, we briefly describe the nonlinear solver and the timestepping strategy employed.

Nonlinear Solver

At each time-step, the nonlinear system of discrete residual equations, i.e.

r(x) = 0

is solved by employing the Newton-Raphson method. Here, x is the vector of primary unknowns. Thus, each physics solver is responsible for assembling the Jacobian matrix J containing the analytical derivatives of the residual vector r with respect to the primary variables. Then, at each Newton iteration \nu, the following linear system is solved

J^{\nu} \delta x^{\nu+1} = -r^{\nu},

where, \delta x^{\nu+1} is the Newton update. This linear system can be solved with a variety of different linear solvers described in Linear Solvers. The Newton update, \delta x^{\nu+1} is then applied to the primary variables:

x^{\nu+1} = x^{\nu} + \delta x^{\nu+1}.

This procedure is repeated until convergence is achieved or until the maximum number of iterations is reached.

Timestepping Strategy

The actual timestep size employed is determined by a combination of several factors. In particular, specific output events may have timestep requirements that force a specific timestep to be used. However, physics solvers do have the possibility of requesting a specific timestep size to the event manager based on their specific requirements. In particular, in case of fast convergence indicated by a small number of Newton iterations, i.e.

\text{numIterations} < \text{dtIncIterLimit} \cdot \text{newtonMaxIter},

the physics solver will require to double the timestep size. On the other hand, if a large number of nonlinear iterations are necessary to find the solution at timestep n

\text{numIterations} > \text{dtCutIterLimit} \cdot \text{newtonMaxIter},

the physics solver will request the next timestep, n+1, to be half the size of timestep n. Here,

Additionally, in case the nonlinear solver fails to converge with the timestep provided by the event manager, the timestep size is cut, i.e.

\text{dt} = \text{timestepCutFactor} \cdot \text{dt},

and the nonlinear loop is repeated with the new timestep size.

Parameters

All parameters defining the behavior of the nonlinear solver and determining the timestep size requested by the physics solver are defined in the NonlinearSolverParameters and are presented in the following table.

Name

Type

Default

Description

allowNonConverged

integer

0

Allow non-converged solution to be accepted. (i.e. exit from the Newton loop without achieving the desired tolerance)

couplingType

geos_NonlinearSolverParameters_CouplingType

FullyImplicit

Type of coupling. Valid options:
* FullyImplicit
* Sequential

lineSearchAction

geos_NonlinearSolverParameters_LineSearchAction

Attempt

How the line search is to be used. Options are:
* None - Do not use line search.
* Attempt - Use line search. Allow exit from line search without achieving smaller residual than starting residual.
* Require - Use line search. If smaller residual than starting resdual is not achieved, cut time step.

lineSearchCutFactor

real64

0.5

Line search cut factor. For instance, a value of 0.5 will result in the effective application of the last solution by a factor of (0.5, 0.25, 0.125, …)

lineSearchInterpolationType

geos_NonlinearSolverParameters_LineSearchInterpolationType

Linear

Strategy to cut the solution update during the line search. Options are:
* Linear
* Parabolic

lineSearchMaxCuts

integer

4

Maximum number of line search cuts.

lineSearchResidualFactor

real64

1

Factor to determine residual increase (recommended values: 1.1 (conservative), 2.0 (relaxed), 10.0 (aggressive)).

lineSearchStartingIteration

integer

0

Iteration when line search starts.

logLevel

integer

0

Log level

maxAllowedResidualNorm

real64

1e+09

Maximum value of residual norm that is allowed in a Newton loop

maxNumConfigurationAttempts

integer

10

Max number of times that the configuration can be changed

maxSubSteps

integer

10

Maximum number of time sub-steps allowed for the solver

maxTimeStepCuts

integer

2

Max number of time step cuts

minNormalizer

real64

1e-12

Value used to make sure that residual normalizers are not too small when computing residual norm.

newtonMaxIter

integer

5

Maximum number of iterations that are allowed in a Newton loop.

newtonMinIter

integer

1

Minimum number of iterations that are required before exiting the Newton loop.

newtonTol

real64

1e-06

The required tolerance in order to exit the Newton iteration loop.

nonlinearAccelerationType

geos_NonlinearSolverParameters_NonlinearAccelerationType

None

Nonlinear acceleration type for sequential solver.

normType

geos_solverBaseKernels_NormType

Linfinity

Norm used by the flow solver to check nonlinear convergence. Valid options:
* Linfinity
* L2

sequentialConvergenceCriterion

geos_NonlinearSolverParameters_SequentialConvergenceCriterion

ResidualNorm

Criterion used to check outer-loop convergence in sequential schemes. Valid options:
* ResidualNorm
* NumberOfNonlinearIterations
* SolutionIncrements

subcycling

integer

0

Flag to decide whether to iterate between sequentially coupled solvers or not.

timeStepCutFactor

real64

0.5

Factor by which the time step will be cut if a timestep cut is required.

timeStepDecreaseFactor

real64

0.5

Factor by which the time step is decreased when the number of Newton iterations is large.

timeStepDecreaseIterLimit

real64

0.7

Fraction of the max Newton iterations above which the solver asks for the time-step to be decreased for the next time step.

timeStepIncreaseFactor

real64

2

Factor by which the time step is increased when the number of Newton iterations is small.

timeStepIncreaseIterLimit

real64

0.4

Fraction of the max Newton iterations below which the solver asks for the time-step to be increased for the next time step.