def transport(Q, dt, u, phi_):
# Define test and trial functions
phi = TrialFunction(Q)
v = TestFunction(Q)
# Constants associated with DG scheme
alpha = StaticConstant(10.0)
mesh = Q.mesh()
h = CellSize(mesh)
n = FacetNormal(mesh)
# Diffusivity constant
kappa = StaticConstant(0.0001)
# Define discrete time derivative operator
Dt = lambda phi: backward_euler(phi, phi_, dt)
a_A = lambda phi, v: advection(phi, v, u, n)
a_D = lambda phi, v: diffusion(phi, v, kappa, alpha, n, h)
# Define form
F = Dt(phi) * v * dx + a_A(phi, v) + a_D(phi, v)
return (lhs(F), rhs(F))
def energy(Q, dt, u, T_):
# Define basis functions
T = TrialFunction(Q)
psi = TestFunction(Q)
# Diffusivity constant
kappa = Constant(1.0)
# Variables associated with DG scheme
alpha = Constant(50.0)
mesh = Q.mesh()
h = CellSize(mesh)
n = FacetNormal(mesh)
# Define discrete time derivative operator
Dt = lambda T: backward_euler(T, T_, dt)
a_A = lambda u, T, psi: advection(T, psi, u, n)
a_D = lambda T, psi: diffusion(T, psi, kappa, alpha, n, h)
# Define form
F = Dt(T)*psi*dx + a_A(u, T, psi) + a_D(T, psi)
return (lhs(F), rhs(F))
def energy(Q, dt, u, T_):
# Define basis functions
T = TrialFunction(Q)
psi = TestFunction(Q)
# Diffusivity constant
kappa = StaticConstant(1.0)
# Variables associated with DG scheme
alpha = StaticConstant(50.0)
mesh = Q.mesh()
h = CellSize(mesh)
n = FacetNormal(mesh)
# Define discrete time derivative operator
Dt = lambda T: backward_euler(T, T_, dt)
a_A = lambda u, T, psi: advection(T, psi, u, n)
a_D = lambda T, psi: diffusion(T, psi, kappa, alpha, n, h)
# Define form
F = Dt(T) * psi * dx + a_A(u, T, psi) + a_D(T, psi)
return (lhs(F), rhs(F))
def transport(Q, dt, u, phi_):
# Define test and trial functions
phi = TrialFunction(Q)
v = TestFunction(Q)
# Constants associated with DG scheme
alpha = StaticConstant(10.0)
mesh = Q.mesh()
h = CellSize(mesh)
n = FacetNormal(mesh)
# Diffusivity constant
kappa = StaticConstant(0.0001)
# Define discrete time derivative operator
Dt = lambda phi: backward_euler(phi, phi_, dt)
a_A = lambda phi, v: advection(phi, v, u, n)
a_D = lambda phi, v: diffusion(phi, v, kappa, alpha, n, h)
# Define form
F = Dt(phi)*v*dx + a_A(phi, v) + a_D(phi, v)
return (lhs(F), rhs(F))
def a_A(u, T, psi):
return advection(T, psi, u, n, theta=0.5)
def a_A(u, T, psi):
return advection(T, psi, u, n, theta=0.5)