How can I define a convection term that includes an independent variable?

[LiuDongDong1989]

My equations are as shown in the figure, and while the equations themselves aren't particularly complex, I'm having trouble figuring out how to define them in FiPy. I've tried some code, but I keep getting errors. How can I define the second term in the second equation and the second term in the third equation?

The code for the first version is as follows：

`def CreatEquation(self):
    #h
    self.R = self.P * numerix.cos(self.slopeAngle) - self.i
    self.eq_h = TransientTerm(var=self.h) + ConvectionTerm(var=self.h, coeff=[self.u]) == self.R

    #u
    self.f = 8 * self.g * self.h * numerix.sin(self.slopeAngle) / (self.u**2)
    self.Sfx = (self.u / numerix.sqrt(8 * self.g / self.f))**2 / self.h
    self.eq_u = (TransientTerm(var=self.u) + ConvectionTerm(var=self.u, coeff=[self.u]) + 
                ConvectionTerm(var=self.h, coeff=[self.g])) == (
                    self.g * (numerix.tan(self.slopeAngle) - self.Sfx) +
                    self.P * numerix.cos(self.slopeAngle) * self.Vm / self.h -
                    self.R * self.u / self.h
                )
    
    #h & u
    self.eqns = self.eq_h & self.eq_u    `

According the comments from jeguyer, the code for the second version is as follows:

   `self.R = self.P * numerix.cos(self.slopeAngle) - self.i
    self.eqh = TransientTerm(var=self.h) + ConvectionTerm(var=self.h, coeff=self.u * [[1]]) == self.R   
    self.f = 8 * self.g * self.h * numerix.sin(self.slopeAngle) / (self.u**2)
    self.Sfx = (self.u / numerix.sqrt(8 * self.g / self.f))**2 / self.h
    self.equ = ( TransientTerm(var=self.u) + 
                ConvectionTerm(var=self.u, coeff=(self.u/2) *[[1]])  + 
                (self.g * self.h.arithmeticFaceValue).divergence ) == (
                self.g * (numerix.tan(self.slopeAngle) - self.Sfx) +
                self.P * numerix.cos(self.slopeAngle) * self.Vm / self.h -
                self.R * self.u /self.h 
                )
    # h & u
    self.eqns = self.eqh & self.equ`

The above code can run but can not convergence. However, if I use the ConvectionTerm(coeff=self.g, var=self.h) - ImplicitSourceTerm(coeff=self.g.grad, var=self.h) to represent the third term of the third equation, the error was:
AttributeError: 'float' object has no attribute 'grad'. Here, the g was gravitational acceleration(9.8 m^2/s)

jeguyer told me that self.g * self.h.arithmeticFaceValue).divergence can not represent the third term.

[guyer]

That's not what I said your equations should look like. Specifically, you have $g\frac{\partial h}{\partial x} = \frac{\partial}{\partial x}\left(g h\right) - h\frac{\partial g}{\partial x}$, which I said could be written as ConvectionTerm(coeff=g, var=h) - ImplicitSourceTerm(coeff=g.grad, var=h). If $g$ is constant, then you can omit the ImplicitSourceTerm. For the ConvectionTerm, the coeff needs to be a vector. If you have defined self.g = 9.8, then you can write ConvectionTerm(coeff=[self.g], var=h). The square brackets around self.g mean that this is a Python list with one item in it; FiPy will treat this as a 1D vector.

[LiuDongDong1989]

This codeConvectionTerm(coeff=[self.g], var=h) is right and can run, but ConvectionTerm(coeff=self.g * [[1]], var=h) can not run. What is the differences between them? And self.g * self.h.arithmeticFaceValue).divergence can not represent the third term?

[guyer]

• If self.g is a float, then self.g * [[1]] is attempting to multiply a float by a list. Python doesn't know how to do that. The reason that self.u * [[1]] works is because self.u is a FiPy CellVariable. A CellVariable "knows" that when it's multiplied with a list, that it should try to interpret that list as a vector. You could get the same behavior with self.g if you wrote

  self.g = fp.Variable(9.8)
  ConvectionTerm(coeff=self.g * [[1]], var=h)

• (self.g * self.h.arithmeticFaceValue).divergence represents $\nabla\cdot(g h)$ (or $\partial(g h)/\partial x$ in 1D). While, when and only when $g$ is constant, this is mathematically equivalent to $g (\partial h/\partial x)$, the expression (self.g * self.h.arithmeticFaceValue).divergence is fully explicit in FiPy and will converge poorly compared to ConvectionTerm(coeff=[self.g], var=h).