def SchemeCentered(u, cst, mult, omega, diff, bc, ret_hmax=False):
"""Discretization of a linear non-divergence form second order PDE
cst + mult u + <omega,grad u>- tr(diff hess(u)) = 0
Second order accurate, centered yet monotone finite differences are used for <omega,grad u>
- bc : boundary conditions.
- ret_hmax : return the largest grid scale for which monotony holds
"""
# Decompose the tensor field
coefs2, offsets = Selling.Decomposition(diff)
# Decompose the vector field
scals = lp.dot_VA(lp.solve_AV(diff, omega), offsets.astype(float))
coefs1 = coefs2 * scals
if ret_hmax: return 2. / norm(scals, ord=np.inf)
# Compute the first and second order finite differences
du = bc.DiffCentered(u, offsets)
d2u = bc.Diff2(u, offsets)
# In interior : cst + mult u + <omega,grad u>- tr(diff hess(u)) = 0
coefs1, coefs2 = (bc.as_field(e) for e in (coefs1, coefs2))
residue = cst + mult * u + lp.dot_VV(coefs1, du) - lp.dot_VV(coefs2, d2u)
# On boundary : u-bc = 0
return np.where(bc.interior, residue, u - bc.grid_values)
def Gradient(u, A, bc, decomp=None):
"""
Approximates grad u(x), using finite differences along the axes of A.
"""
coefs, offsets = Selling.Decomposition(A) if decomp is None else decomp
du = bc.DiffCentered(u, offsets)
AGrad = lp.dot_AV(offsets.astype(float),
(coefs * du)) # Approximates A * grad u
return lp.solve_AV(A, AGrad) # Approximates A^{-1} (A * grad u) = grad u
