Tokamak axisymmetric transport
Simulations of transport in axisymmetric tokamak geometries, with cross-field diffusion and interaction of plasma with neutral gas.
Finding steady state solutions
These models can be run as a time-dependent problem, for example to study power transients, but the primary application is to finding steady-state solutions.
Backward Euler solver
This solver uses PETSc to solve the nonlinear system of equations, with a Backward Euler timestep to improve the condition number. There are many choices of algorithm and settings, so the following are guidelines and may not be optimal for all cases.
[solver]
type = beuler # Backward Euler steady-state solver
snes_type = newtonls # Nonlinear solver
ksp_type = gmres # Linear solver
max_nonlinear_iterations = 10
pc_type = hypre # Preconditioner type
pc_hypre_type = euclid # Hypre preconditioner type
lag_jacobian = 500 # Iterations between jacobian recalculations
atol = 1e-7 # Absolute tolerance
rtol = 1e-5 # Relative tolerance
PETSc can print quite extensive performance diagnostics. These can be enabled by putting in the BOUT.inp options file:
[petsc]
log_view = true
This section can also be used to set other PETSc flags, just omitting
the leading - from the PETSc option.
cvode solver
CVODE is primarily intended for high-accuracy time integration, rather than finding steady-state solutions, but can be effective and quite robust. It tends to struggle at high order, so here we limit it to a maximum of 3rd order:
[solver]
type = cvode
use_precon = true # Use the user-provided preconditioner
mxstep = 1e5
mxorder = 3 # Limit to 3rd order
atol = 1e-12
rtol = 1e-5
Here use_precon = true tells the solver to use the Hermes-3
preconditioners, which are implemented in some components. This
includes preconditioning of parallel heat conduction, and of
cross-field diffusion of neutrals.
Mesh interpolation
A useful strategy is to start with a low resolution grid, run until close to steady-state, then interpolate the solution onto a finer mesh and restart. This process can be repeated as a kind of simplified multigrid method.