Is the package able to deal with non-constant coefficients?

[ItayXD]

Hi,
I didn't see it mentioned anywhere and it doesn't appear in any of the examples.
Can I define a PDE of the form:

where D(x) could be a polynomial of x for example

Thanks
Itay

[david-zwicker]

This is certainly possible. The simplest way would be to use the PDE class,

from pde import PDE
eq = PDE({'P': '(P - P**2) * laplace(P)'})

where I used D(x) = x - x**2 as a concrete example. Alternatively, you may implement your own pde class.

See pde_custom_expression.py and pde_custom_class.py in the examples directory for respective implementations.

[ItayXD]

I may be missing something, but it seems that you actually used D(P)=P - P**2 rather than D(x) = x - x**2, meaning the coefficient depends on the variable being solved for and not x (which P is being differentiated with respect to)

[david-zwicker]

Ohh, I wasn't reading your question carefully enough – sorry about this. The current implementation of the PDE class does not allow spatially varying coefficients. However, it is not too difficult to achieve this with a custom class:

import pde


class HeterogeneousDiffusionPDE(pde.PDEBase):

    def evolution_rate(self, state, t=0):
        """ implement the python version of the evolution equation """
        assert state.grid.dim == 1  # implementation only works for 1d problems
        cell_coords = state.grid.cell_coords[:, 0]  # this extracts the x-coordinates of all cells
        coeff = (cell_coords - cell_coords**2)  # this calculates the pre-factor
        return state.laplace('natural') * coeff


grid = pde.CartesianGrid([[0, 1]], 32)  # generate grid
state = pde.ScalarField.random_normal(grid, 0.1, 0.9)  # generate initial condition

eq = HeterogeneousDiffusionPDE()  # define the pde
result = eq.solve(state, t_range=1, dt=1e-4)
result.plot()

A numba-compiled implementation to accelerate the computation can be created similarly. I'll think about extending the PDE class, but it might be a bit challenging and thus not happen soon. However, pull requests in this direction are welcome!

[david-zwicker]

I took you suggestion as a challenge and implemented an extended version of the PDE class that can deal with spatially dependent coefficients. The implementation is not very tested, yet, but it seems to work. I added an example that displays how it works, but it really comes down to just using the spatial coordinates as variables in the expressions.

[ItayXD]

Thanks! Looks good