colocated_varrho_symbol Namespace Reference

Functions
	model (N, rho, mu, walls=False)
	step_matrix (m, dt, kap, scheme)
	radius (T)
	density (N, kind, ratio, seed=1)
	main ()

Detailed Description

Step-operator model of the collocated variable-density step (doc/collocated_varrho_forces.md,
Appendix A) -- an INSTRUMENT, not a gate.

A dense, exact-solve, 2-D, h = 1 model of one step (rho0 = 1), unknowns (u, v, P). The step matrix
T maps (u^n, v^n, P^n) -> (u^{n+1}, v^{n+1}, P^{n+1}) exactly (dense inverse for the implicit
momentum operator, pseudo-inverse for the pressure Poisson), so its spectral radius is the per-step
growth factor of the scheme. Three schemes:

  "V8"  the retired WO-T step: the pressure and every force as a face acceleration added AFTER the
        implicit viscous solve (Basilisk centered.h). Non-incremental; with the rotational update
        (kappa = mu) its pressure obeys P^{n+1} = -4 kappa dt S P^n / rho -> radius 8 mu dt (2-D).
  "D2"  the predictor form with the ARITHMETIC centre->face map: not an adjoint pair, radius > 1
        at high contrast.
  "D3"  the chosen scheme: the predictor form with the MOMENTUM-weighted centre->face map
        (rho_L u_L + rho_R u_R)/(rho_L + rho_R), so M Gamma = -C^T and the radius is exactly 1.

Run: OMP_NUM_THREADS=8 python tests/study/colocated_varrho_symbol.py
Expected (the design note): V8 at dt 0.2: 1.6 = 8 mu dt; D2 at ratio 1e4 slightly above 1 at
dt 3 and 10 (the note quotes 1.0136 / 1.0145 for its shape); D3: 1.00000 everywhere.

Function Documentation

model()

colocated_varrho_symbol.model	(		N,
			rho,
			mu,
			walls = False
	)

Definition at line 28 of file colocated_varrho_symbol.py.

Referenced by main().

step_matrix()

colocated_varrho_symbol.step_matrix	(		m,
			dt,
			kap,
			scheme
	)

Definition at line 59 of file colocated_varrho_symbol.py.

Referenced by main().

radius()

colocated_varrho_symbol.radius

(

T

)

Definition at line 85 of file colocated_varrho_symbol.py.

Referenced by main().

density()

colocated_varrho_symbol.density	(		N,
			kind,
			ratio,
			seed = 1
	)

Cell density field (x fastest, index x + N*y).

Definition at line 90 of file colocated_varrho_symbol.py.

Referenced by main().

main()

colocated_varrho_symbol.main

(

)

Definition at line 109 of file colocated_varrho_symbol.py.

References density(), main(), model(), radius(), and step_matrix().

Referenced by main().