SystemModel

SystemModel#

A SystemModel freezes the model into operator slots of one balance law:

$$ M,\partial_t Q + \nabla\cdot\big(F + P\big) + \sum_d B_d,\partial_d Q = S + \nabla\cdot\big(A : \nabla Q\big). $$

What you write: usually nothing but SystemModel.from_model(model) and the initial conditions. The solvers read the slots; they never inspect the model.

slot	shape	treatment
`flux` $F$	`(n_eq, n_dim)`	explicit
`hydrostatic_pressure` $P$	`(n_eq, n_dim)`	explicit (well-balanced with the NCP)
`nonconservative_matrix` $B$	`(n_eq, n_state, n_dim)`	explicit, path-conservative
`source` $S$	`(n_eq, 1)`	implicit slot: implicit under IMEX, else in the RK step
`source_explicit`	`(n_eq,)`	always explicit
`diffusion_matrix` $A$	`(n_eq, n_state, n_dim, n_dim)`	implicit slot, as `source`
`diffusion_matrix_explicit`	same	always explicit
`eigenvalues`	`(n_eq, 1)` or `None`	`None`: numerical wave speed (power iteration)
`mass_matrix` $M$	`(n_eq, n_state)`

Spatial derivatives the operators need are registered as aux variables (dhdx, …) and filled by the solver with least-squares gradients.

The C1 bump of the quick start, inspected (docs/book/examples/inspect_bump.py):

state       ['b', 'h', 'q_x_0', 'q_y_0']
aux_state   ['dq_x0dx', 'dhdx', 'dq_y0dy', 'dhdy', 'dbdx', 'dbdy', 'hinv']
parameters  {'g': 9.81, 'rho': 1.0, 'nu': 0.0, 'lambda_s': 0.0, 'e_x': 0.0, 'n': 0.0}
flux                       shape (4, 2), nonzero 4
hydrostatic_pressure       shape (4, 2), nonzero 4
nonconservative_matrix     shape (4, 4, 2), nonzero 4
source                     shape (4, 1), nonzero 4
source_explicit            None
diffusion_matrix           None
diffusion_matrix_explicit  None
eigenvalues                None

you set	how
initial state	`sm.initial_conditions = IC.UserFunction(function=f)` (`zoomy_core.model.initial_conditions`; `f(x)` returns the state at a point)
initial aux	`sm.aux_initial_conditions`
parameter values	`sm.parameter_values`
pressure split (Chorin)	`VAM.chorin_split(dt, system_model=sm)` gives `SM_pred`, `SM_press`, `SM_corr`

Connects to: NumericalSystemModel.from_system_model(sm, ...) (NumericalSystemModel). sm.describe() renders the slots.

API: zoomy_core.systemmodel.