def solve_heat_equation(k, time_stepping_method):
"""
Solve the heat equation on a hard-coded mesh with a hard-coded initial and boundary conditions
:param k: Thermal conductivity
:param time_stepping_method: Time stepping method. Can be one of ["forward_euler", "backward_euler", "trapezoidal"]
"""
mesh, boundary = setup_geometry()
# Exact solution (Gauss curve)
ue = Expression("exp(-(x[0]*x[0]+x[1]*x[1])/(4*a*t))/(4*pi*a*t)",
a=k,
t=1e-7,
domain=mesh,
degree=1)
# Polynomial degree
r = 1
# Setup FEM function space
V = FunctionSpace(mesh, "CG", r)
# Create boundary condition
bc = DirichletBC(V, ue, boundary)
# Setup FEM functions
v = TestFunction(V)
u = Function(V)
# Time parameters
time_step = 0.001
t_start, t_end = 0.0, 20.0
# Time stepping
t = t_start
if time_stepping_method == "forward_euler":
theta = 0.0
if time_stepping_method == "backward_euler":
theta = 1.0
if time_stepping_method == "trapezoidal":
theta = 0.5
u0 = ue
step = 0
while t < t_end:
# Intermediate value for u (depending on the chosen time stepping method)
um = (1.0 - theta) * u0 + theta * u
# Weak form of the heat equation
a = (u - u0) / time_step * v * dx + k * inner(grad(um), grad(v)) * dx
# Solve the heat equation (one time step)
solve(a == 0, u, bc)
# Advance time in exact solution
t += time_step
ue.t = t
if step % 100 == 0:
# Compute error in L2 norm
error_L2 = errornorm(ue, u, 'L2')
# or equivalently
# sqrt(assemble((ue - u) * (ue - u) * dx))
# Compute norm of exact solution
nue = norm(ue)
# Print relative error
print("Relative error = {}".format(error_L2 / nue))
# Shift to next time step
u0 = project(u, V)
step += 1
def solve_wave_equation(a, symmetric=True):
"""
Solve the wave equation on a hard-coded mesh with a hard-coded initial and boundary conditions
:param a: Wave propagation factor
:param symmetric: Whether or not the problem is symmetric
"""
mesh, boundary = setup_geometry()
# Exact solution
if symmetric:
ue = Expression(
"(1-pow(a*t-x[0],2))*exp(-pow(a*t-x[0],2)) + (1-pow(a*t+x[0],2))*exp(-pow(a*t+x[0],2))",
a=a,
t=0,
domain=mesh,
degree=2)
ve = Expression(
"2*a*(a*t-x[0])*(pow(a*t-x[0],2)-2)*exp(-pow(a*t-x[0],2))"
"+ 2*a*(a*t+x[0])*(pow(a*t+x[0],2)-2)*exp(-pow(a*t+x[0],2))",
a=a,
t=0,
domain=mesh,
degree=2)
else:
ue = Expression("(1-pow(a*t+x[0],2))*exp(-pow(a*t+x[0],2))",
a=a,
t=0,
domain=mesh,
degree=2)
ve = Expression(
"2*a*(a*t+x[0])*(pow(a*t+x[0],2)-2)*exp(-pow(a*t+x[0],2))",
a=a,
t=0,
domain=mesh,
degree=2)
# Polynomial degree
r = 1
# Setup FEM function spaces
Q = FunctionSpace(mesh, "CG", r)
W = VectorFunctionSpace(mesh, "CG", r, dim=2)
# Create boundary conditions
bcu = DirichletBC(W.sub(0), ue, boundary)
bcv = DirichletBC(W.sub(1), ve, boundary)
bcs = [bcu, bcv]
# Setup FEM functions
p, q = TestFunctions(W)
w = Function(W)
u, v = w[0], w[1]
# Time parameters
time_step = 0.05
t_start, t_end = 0.0, 5.0
# Time stepping
t = t_start
u0 = ue
v0 = ve
step = 0
while t < t_end:
# Weak form of the wave equation
um = 0.5 * (u + u0)
vm = 0.5 * (v + v0)
a1 = (u - u0) / time_step * p * dx - vm * p * dx
a2 = (v - v0) / time_step * q * dx + a**2 * inner(grad(um),
grad(q)) * dx
# Solve the wave equation (one time step)
solve(a1 + a2 == 0, w, bcs)
# Advance time in exact solution
t += time_step
ue.t = t
ve.t = t
if step % 10 == 0:
# Plot solution at current time step
fig = plt.figure()
plot(u, fig=fig)
plt.show()
# Compute max error at vertices
vertex_values_ue = ue.compute_vertex_values(mesh)
vertex_values_w = w.compute_vertex_values(mesh)
vertex_values_u = np.split(vertex_values_w, 2)[0]
error_max = np.max(np.abs(vertex_values_ue - vertex_values_u))
# Print error
print(error_max)
# Shift to next time step
u0 = project(u, Q)
v0 = project(v, Q)
step += 1
def solve_heat_equation(k):
"""
Solve the heat equation on a hard-coded mesh with a hard-coded initial and boundary conditions
:param k: Thermal conductivity
"""
mesh, boundary = setup_geometry()
# Exact solution (Gauss curve)
ue = Expression("exp(-(x[0]*x[0]+x[1]*x[1])/(4*a*t))/(4*pi*a*t)",
a=k,
t=1e-7,
domain=mesh,
degree=2)
# Polynomial degree
r = 1
# Setup FEM function space
V = FunctionSpace(mesh, "CG", r)
# Create boundary condition
bc = DirichletBC(V, ue, boundary)
# Setup FEM functions
v = TestFunction(V)
u = Function(V)
# Time parameters
time_step = 0.5
t_start, t_end = 0.0, 20.0
# Time stepping
t = t_start
u0 = ue
step = 0
while t < t_end:
# Weak form of the heat equation
a = (u - u0) / time_step * v * dx + k * inner(grad(u), grad(v)) * dx
# Solve the heat equation (one time step)
solve(a == 0, u, bc)
# Advance time in exact solution
t += time_step
ue.t = t
if step % 5 == 0:
# Plot solution at current time step
fig = plt.figure()
plot(u, fig=fig)
plt.show()
# Compute error in L2 norm
error_L2 = errornorm(ue, u, 'L2')
# or equivalently
# sqrt(assemble((ue - u) * (ue - u) * dx))
# Print error
print(error_L2)
# Shift to next time step
u0 = project(u, V)
step += 1