def crank_nicolson(
model: Model, gfu_0: List[ngs.GridFunction], U: List[ProxyFunction],
V: List[ProxyFunction],
dt: List[ngs.Parameter]) -> Tuple[ngs.BilinearForm, ngs.LinearForm]:
"""
Crank Nicolson (trapezoidal rule) time integration scheme.
This function constructs the final bilinear and linear forms for the time integration scheme by adding the
necessary time-dependent terms to the model's stationary terms.
The returned bilinear and linear forms have NOT been assembled.
Args:
model: The model to solve.
gfu_0: List of the solutions of previous time steps ordered from most recent to oldest.
U: List of trial functions for the model.
V: List of test (weighting) functions.
dt: List of timestep sizes ordered from most recent to oldest.
Returns:
a: The final bilinear form (as a ngs.BilinearForm but not assembled).
L: The final linear form (as a ngs.LinearForm but not assembled).
"""
# Separate out the various components of each gridfunction solution to a previous timestep.
# gfu_lst = [[component 0, component 1...] at t^n, [component 0, component 1...] at t^n-1, ...]
gfu_lst = []
for i in range(len(gfu_0)):
if (len(gfu_0[i].components) > 0):
gfu_lst.append([
gfu_0[i].components[j] for j in range(len(gfu_0[i].components))
])
else:
gfu_lst.append([gfu_0[i]])
# Construct the bilinear form
a = ngs.BilinearForm(model.fes)
a += 0.5 * model.construct_bilinear(U, V, dt[0])
# Construct the linear form
L = ngs.LinearForm(model.fes)
L += model.construct_linear(V, gfu_lst[0], dt[0])
L += -0.5 * model.construct_bilinear(gfu_lst[0], V, dt[0], True)
a_dt, L_dt = model.time_derivative_terms(gfu_lst, 'crank nicolson')
# When adding the time discretization term, multiply by the phase field if using the diffuse interface method.
if model.DIM:
a_dt *= model.DIM_solver.phi_gfu
L_dt *= model.DIM_solver.phi_gfu
a += a_dt * dx
L += L_dt * dx
return a, L
def adaptive_IMEX_pred(
model: Model, gfu_0: List[ngs.GridFunction], U: List[ProxyFunction],
V: List[ProxyFunction],
dt: List[ngs.Parameter]) -> Tuple[ngs.BilinearForm, ngs.LinearForm]:
"""
Predictor for the adaptive time-stepping IMEX time integration scheme.
This function constructs the final bilinear and linear forms for the time integration scheme by adding the
necessary time-dependent terms to the model's stationary terms.
The returned bilinear and linear forms have NOT been assembled.
Args:
model: The model to solve.
gfu_0: List of the solutions of previous time steps ordered from most recent to oldest.
U: List of trial functions for the model.
V: List of test (weighting) functions.
dt: List of timestep sizes ordered from most recent to oldest.
Returns:
a: The final bilinear form (as a ngs.BilinearForm but not assembled).
L: The final linear form (as a ngs.LinearForm but not assembled).
"""
# Separate out the various components of each gridfunction solution to a previous timestep.
# gfu_lst = [[component 0, component 1...] at t^n, [component 0, component 1...] at t^n-1, ...]
gfu_lst = []
for i in range(len(gfu_0)):
if (len(gfu_0[i].components) > 0):
gfu_lst.append([
gfu_0[i].components[j] for j in range(len(gfu_0[i].components))
])
else:
gfu_lst.append([gfu_0[i]])
# Operators specific to this time integration scheme.
w = dt[0] / dt[1]
E = model.construct_gfu().components[0]
E_expr = (1.0 + w) * gfu_lst[0][0] - w * gfu_lst[1][0]
E.Set(E_expr)
# Construct the bilinear form
a = ngs.BilinearForm(model.fes)
a += model.construct_bilinear(U, V, dt[0])
# Construct the linear form
L = ngs.LinearForm(model.fes)
L += model.construct_linear(
V, [gfu_lst[0][0], 0.0],
dt[0]) # TODO: Why doesn't using E work? Numerical error?
a_dt, L_dt = model.time_derivative_terms(gfu_lst, 'crank nicolson')
# When adding the time discretization term, multiply by the phase field if using the diffuse interface method.
if model.DIM:
a_dt *= model.DIM_solver.phi_gfu
L_dt *= model.DIM_solver.phi_gfu
a += a_dt * dx
L += L_dt * dx
return a, L