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

Question

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 PDE Equations

Error:
--> 250     assert len(id1) == len(id2) == len(vector)
    252     temp = sp.csr_matrix((vector, (id1, id2)), self.matrix.shape)
    254     self.matrix = self.matrix + temp

AssertionError: 

    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

When I set the coeff to 1, I don't encounter any errors, but that doesn't represent the equation correctly. How can I define the second term in the second equation and the second term in the third equation without causing errors?

Answer 1

A couple of things:

• [self.u] is not a vector field; it's a Python list with a scalar field in it. Instead, write self.u * [[1]].
• and aren't in canonical form for convection terms.
  • , so you could write ConvectionTerm(coeff=u/2, var=u).
  • , so you could write ConvectionTerm(coeff=g, var=h) - ImplicitSourceTerm(coeff=g.grad, var=h).

Generally speaking, 1D equations are an excellent way to get really confused about how to represent things in FiPy (and in general, IMO). It's impossible to tell which derivatives are physically vector and physically scalar.