problems in solving 2D Allen Cahn equation

[刘子源(Liu Ziyuan)]

Hi Mikael and sorry to bother you again, I try solving 2D Allen Cahn equation but the solution shenfun returned seems not correct, since generally the solution of ACE will converge to 1 and -1 after sufficient time. Below is the toy experiment.

import matplotlib
matplotlib.use('TkAgg')
from time import time
import numpy as np
import matplotlib.pyplot as plt
import sympy as sp
from shenfun import *
from copy import deepcopy

# Use sympy to set up initial condition
x, y = sp.symbols("x, y")
u0 = sp.sin(sp.pi*x)*sp.sin(sp.pi*y)/2

# parameters for AC equation
eps = 0.02

Nx = 100 + 1

CX = FunctionSpace(Nx, 'C', quad='GL', bc={'left':('D',0), 'right':('D',0)})
CY = FunctionSpace(Nx, 'C', quad='GL', bc={'left':('D',0), 'right':('D',0)})
C = TensorProductSpace(comm, (CX, CY))

u = TrialFunction(C)
v = TestFunction(C)

def LinearRHS(self, u, **par):
    return eps * div(grad(u)) #+ u

def NonlinearRHS(self, u, u_hat, rhs, **params):
    # u-u**3
    rhs.fill(0)
    ub = u_hat.backward(padding_factor=2)
    ub[:] = -ub**3 + ub
    rhs = ub.forward(rhs)
    return rhs

u_ = Array(C, buffer=u0)
u_hat = Function(C)
u_hat = u_.forward(u_hat)

def update(self, u, u_hat, t, tstep, **kw):
    if tstep % 100 == 0:
        print(tstep)
        print(np.max(np.abs(u_hat.backward())))
        # fig, ax = plt.subplots()
        # X, Y = np.meshgrid(*C.mesh())
        # cs = ax.contourf(X, Y, u_hat.backward(), cmap=plt.get_cmap('Spectral'))
        # # ax.plot_surface(-XCGL, -YCGL, u0.backward())
        # cbar = fig.colorbar(cs)
        # plt.show()
        X, Y = np.meshgrid(*C.mesh())
        fig = plt.figure()
        plt.pcolor(X, Y, u_hat.backward(), shading='auto', cmap="jet")
        plt.colorbar()
        plt.show()

#dt = 1/Nx
dt = 0.001
end_time = 10
#integrator = BackwardEuler(C, L=LinearRHS, N=NonlinearRHS, update=update)
integrator = IRK3(C, L=LinearRHS, N=NonlinearRHS, update=update)
integrator.setup(dt)
u_hat = integrator.solve(u_, u_hat, dt, (0, end_time))

[Mikael Mortensen]

Hi
As far as I can tell you solve a problem with homogeneous Dirichlet boundary conditions and the solution should then converge to 0 and not +1 or -1?

[Mikael Mortensen]

There is an error in the NonlinearRHS, that needs to be a scalar product and not a complete forward transform. The following should work:

def NonlinearRHS(self, u, u_hat, rhs, **params):
    # u-u**3
    rhs.fill(0)
    ub = u_hat.backward(padding_factor=2)
    uh = (-ub**3 + ub).forward()
    rhs = inner(v, uh, output_array=rhs)
    return rhs