def parabolic_equation_solver_test(self, NS, NT):
"""
u_t = \Delta u + w*u
w = 1
u(x, 0) = 1
周期边界条件
"""
from fealpy.timeintegratoralg.timeline_new import UniformTimeLine
box = np.array([
[2*np.pi, 0],
[0, 2*np.pi]])
space = FourierSpace(box, NS)
timeline = UniformTimeLine(0, 1, NT)
NL = timeline.number_of_time_levels()
q = space.function(dim=NL)
w = space.function()
w[:] = 1
q[0] = 1
k, k2 = space.reciprocal_lattice(return_square=True)
dt = timeline.current_time_step_length()
E0 = np.exp(-dt/2*w)
E1 = np.exp(-dt*k2)
for i in range(1, NL):
q0 = q[i-1]
q1 = np.fft.fftn(E0*q0)
q1 *= E1
q[i] = np.fft.ifftn(q1).real
q[i] *= E0
print(q)
def parabolic_equation_solver_uu_test(self, NS, NT):
"""
u_t = \Delta u + u^2
w = 1
u(x, 0) = 1
周期边界条件
Semi-implicit method
The discrete scheme: (time step size is dt)
\hat{u}^{n+1} = ( \hat{u}^{n} + dt \hat{(u^{n})^{2}} ) / ( I + dt k^2 )
"""
from fealpy.timeintegratoralg.timeline import UniformTimeLine
box = np.array([
[2*np.pi, 0],
[0, 2*np.pi]])
space = FourierSpace(box, NS)
timeline = UniformTimeLine(0, 1, NT)
NL = timeline.number_of_time_levels()
u = space.function(dim=NL)
u[0] = 1
k, k2 = space.reciprocal_lattice(return_square=True)
dt = timeline.current_time_step_length()
A = 1./(1 + dt*k2)
for i in range(1, NL):
u0 = u[i-1]
u1 = np.fft.fftn(u0)
uf = np.fft.fftn(u0*u0)
u2 = A * (u1 + dt*uf)
u[i] = np.fft.ifftn(u2).real
print(u)
