#include "petscfv.h" PetscErrorCode PetscLimiterLimit(PetscLimiter lim, PetscReal flim, PetscReal *phi)
| lim | - The PetscLimiter | |
| flim | - The input field |
| phi | - The limited field |
Note: Limiters given in symmetric form following Berger, Aftosmis, and Murman 2005
The classical flux-limited formulation is psi(r) where
r = (u[0] - u[-1]) / (u[1] - u[0])
The second order TVD region is bounded by
psi_minmod(r) = min(r,1) and psi_superbee(r) = min(2, 2r, max(1,r))
where all limiters are implicitly clipped to be non-negative. A more convenient slope-limited form is psi(r) =
phi(r)(r+1)/2 in which the reconstructed interface values are
u(v) = u[0] + phi(r) (grad u)[0] v
where v is the vector from centroid to quadrature point. In these variables, the usual limiters become
phi_minmod(r) = 2 min(1/(1+r),r/(1+r)) phi_superbee(r) = 2 min(2/(1+r), 2r/(1+r), max(1,r)/(1+r))
For a nicer symmetric formulation, rewrite in terms of
f = (u[0] - u[-1]) / (u[1] - u[-1])
where r(f) = f/(1-f). Not that r(1-f) = (1-f)/f = 1/r(f) so the symmetry condition
phi(r) = phi(1/r)
becomes
w(f) = w(1-f).
The limiters below implement this final form w(f). The reference methods are
w_minmod(f) = 2 min(f,(1-f)) w_superbee(r) = 4 min((1-f), f)