Solvers

Runge Kutta

The Runge Kutta solver is an explicit solver. It is configured by setting config.solver in job.py to an instance of RungeKutta. For example:

config.solver = RungeKutta(
    method="ssp-rk3",
    cfl = 0.5,
    max_step = 10000,
    max_time = 1e-3,
    plot_every_n_steps = -1,
    plot_frequency = 1e-4,
    print_frequency = 500,
    residual_frequency = 1e-5,
    residuals_every_n_steps = 20,
)

Each option is optional, default values will be used if they are not provided.

Each option is described below.

method

The specific Runge-Kutta scheme to use. The options are:

• euler
• midpoint
• rk3
• ssp-rk3
• rk4

Type: str
Default: ssp-rk3

butcher_tableau

Instead of using one of the existing Runge-Kutta methods, you may provide the Butcher tableau for any explicit Runge Kutta method. The tableau is specified as a dictionary of the b, b, and c values of the Butcher tableau. Only the a values below the diagonal should are included, and the first c value, which is always 0 for explicit Runge-Kutta methods is not included either. For example, the classic RK4 method can be specified as:

butcher_tableau = {
  "a": [[0.5], [0, 0.5], [0, 0, 1]],
  "b": [1/6, 1/3, 1/3, 1/6],
  "c": [0.5, 0.5, 1.0]
}

It is an error to provide both method and butcher_tableau to the RungeKutta constructor.

cfl

The cfl value to dictate how big of a step to take

Type: float
Default: 0.5

max_step

The maximum number of steps to take. This can be helps stop a job from spinning its wheels going nowhere.

Type: int
Default: 100

print_frequency

The number of steps between printing the progress of the simulation to the terminal

Type: float
Default: 1e-5

residuals_every_n_steps

The number of steps between writing norms of the residuals. Setting to -1 disables this.

Type: int
Default: 20

residual_frequency

The time interval between writing norms of residuals to file Setting to -1 disables this.

Type: float
Default: 1e-5

max_time

The maximum time to simulate.

Type: float
Default: 1e-3

plot_every_n_steps

Write a snapshot at intervals a fixed number of steps apart. This can be useful for debugging a simulation that is struggling to start. Setting it to -1 disables this behaviour.

Type: int
Default: -1

plot_frequency

Write a snapshot at fixed intervals of simulation time.

Type: float
Default: 1e-3

Steady State

The steady-state solver uses the Jacobian-Free Newton-Krylov method to accelerate convergence to steady-state. It is configured by setting config.solver in job.py to an instance of SteadyState. For example:

config.solver = SteadyState(
  cfl=ResidualBasedCfl(growth_threshold=0.9, power=0.85, start_cfl=0.5),
  max_steps=1000,
  print_frequency=20,
  plot_frequency=100,
  diagnostics_frequency=1,
  tolerance=1e-10,
  linear_solver=Gmres(tol=1e-4, max_iters=50)
)

Each option is optional, default values will be used if they are not provided.

Each option is described below.

cfl

The cfl to use for the simulation.

Type: float | CflSchedule Default: 0.5

max_steps

The maximum number of steps to take

Type: int
Default 1000

print_frequency

The number of steps between printing progress to the screen

Type: int
Default: 10

plot_frequency

The number of steps between writing the flow solution to disk

Type: int
Default: 10

diagnostics_frequency

The number of steps between writing diagnostics. Diagnostics are written in the log folder.

Type: int
Default: 1

tolerance

The drop in the relative global residual required for convergence

Type: float
Default: 1e-5

linear_solver

The linear to use for each non-linear step

Type: LinearSolver