使用FiPy解决圆柱坐标系中的扩散方程不正确

Question

使用FiPy对圆柱几何中的扩散方程的稳态解决方案与从另一软件（例如，Mathematica）获得的解决方案相当不同。

等式是：

$ 0 = \ frac {1} {r} \ frac {d} {dr} \ left（\ frac {r} {T ^ {1/2}} \ frac {dT} {dr} \ right）+ cte * T ^ {3/2} $

这意味着，通过使用CylindricalGrid1D网格，我们可以将等式写为：

mesh = CylindricalGrid1D(nr=100, dr=0.01, origin=0.0)
T = CellVariable(name='temperature', mesh=mesh, hasOld=True)
r = mesh.cellCenters()

#BC's
T.constrain(0., mesh.facesRight)
T.faceGrad.constrain(0.,mesh.facesLeft)

#initial temperature profile
T.setValue( 1-r**2)

eq = 0 == DiffusionTerm( coeff=T**(-1/2), var=T) + 20*ImplicitSourceTerm(coeff=T**(1/2), var=T)

viewer = Viewer(vars=T)
eq.solve()

viewer.plot()
raw_input(" Press <enter> to proceed...")

在这里我设置了cte = 20，但问题仍然是这个值。我得到左边的解决方案，而Mathematica给出的解决方案是右边的解决方案：

plots

然后我尝试按照这样的非线性方程推荐扫描。所以，我没有使用eq.solve()，而是：

current_residual = 1.0e100
desired_residual = 1e-5

while current_residual > desired_residual:

    current_residual = eq.sweep()
    T.updateOld()

但我得到错误：

/home/antonio/.local/lib/python2.7/site-packages/fipy/solvers/scipy/linearLUSolver.py:66: RuntimeWarning: overflow encountered in square
  error0 = numerix.sqrt(numerix.sum((L * x - b)**2))
/home/antonio/.local/lib/python2.7/site-packages/fipy/solvers/scipy/linearLUSolver.py:71: RuntimeWarning: overflow encountered in square
  if (numerix.sqrt(numerix.sum(errorVector**2)) / error0)  <= self.tolerance:
/home/antonio/.local/lib/python2.7/site-packages/fipy/solvers/scipy/linearLUSolver.py:71: RuntimeWarning: invalid value encountered in double_scalars
  if (numerix.sqrt(numerix.sum(errorVector**2)) / error0)  <= self.tolerance:
/home/antonio/.local/lib/python2.7/site-packages/fipy/tools/numerix.py:966: RuntimeWarning: overflow encountered in square
  return sqrt(add.reduce(arr**2))
/home/antonio/.local/lib/python2.7/site-packages/fipy/solvers/scipy/linearLUSolver.py:58: RuntimeWarning: overflow encountered in multiply
  b = b * (1 / maxdiag)
Traceback (most recent call last):
  File "stack.py", line 26, in <module>
    current_residual = eq.sweep()
  File "/home/antonio/.local/lib/python2.7/site-packages/fipy/terms/term.py", line 254, in sweep
    solver._solve()
  File "/home/antonio/.local/lib/python2.7/site-packages/fipy/solvers/scipy/scipySolver.py", line 61, in _solve
    self.var[:] = numerix.reshape(self._solve_(self.matrix, self.var.ravel(), numerix.array(self.RHSvector)), self.var.shape)   
  File "/home/antonio/.local/lib/python2.7/site-packages/fipy/solvers/scipy/linearLUSolver.py", line 64, in _solve_
    permc_spec=3)
  File "/usr/lib/python2.7/dist-packages/scipy/sparse/linalg/dsolve/linsolve.py", line 257, in splu
    ilu=False, options=_options)
RuntimeError: Factor is exactly singular

最后，我使用变量V = T ^ {1/2}以等效形式重写初始方程。很容易看出，随着V，等式变为

$ 0 = \ frac {1} {r} \ frac {d} {dr} \ left（r \ frac {dV} {dr} \ right）+ \ frac {cte} {2} V ^ 3 $

所以我然后使用了代码：

mesh = CylindricalGrid1D(nr=100, dr=0.01, origin=0.0)
V = CellVariable(name='V', mesh=mesh, hasOld = True)
r = mesh.cellCenters()

#BC's
V.constrain(0., mesh.facesRight)
V.faceGrad.constrain(0.,mesh.facesLeft)

#initial V profile
V.setValue( 1-r**2)

eqV = 0 == DiffusionTerm( coeff=1., var=V) + 20*0.5*ImplicitSourceTerm(coeff=V*V, var=V)

T = V*V
viewer = Viewer(vars=T)

eqV.solve()

viewer.plot()
raw_input(" Press <enter> to proceed...")

获得的轮廓非常相似，但y轴的值与第一个FiPy解决方案或Mathematica解决方案不同！扫地给出了与以前相同的错误。

Answer 1

我不相信这个问题除了T = 0之外还有任何解决方案。此外，该解决方案似乎对初始条件和/或cte的不同值不稳定。考虑到当t = 0时，T形式的方程将具有无限的扩散性，这种不稳定性并不完全令人惊讶。

我怀疑Mathematica在你的第一组数字中大致正在做FiPy正在做的事情，这是为了显示这个非线性问题的第一次扫描。这不是答案;只是第一个猜测。但是，无论是分析还是数字，我都不知道用Mma解决偏微分方程。

在你的V解决方案之后一次扫描后的情节看起来不同，顺便说一下，因为你没有调整初始条件。它应该是：

V.setValue( numerix.sqrt(1-r**2))