Is gradient[0] equivalent to d_dx?

[adamg-cci]

I am building a system of equations that uses the spatial derivative in many of the equations (d_dx). I see that in example 2.12 the string representation of this with sympy is d_dx. The equivalent example with a custom class, example 2.20 uses gradient()[0]. It seems to me that this is functionally equivalent to d_dx, but I wanted to confirm through the documentation and data structures that this is the case. However I wasn't able to get complete clarity on this. I can't imagine that something this fundamental wouldn't be supported by the package, so I assume the use of gradient is intended. Just wanted to get some ultimate clarity on this. Thanks!
Adam

[adamg-cci]

The formulation of my equations does not use gradient or laplace at all, so I am also wondering if this is actually an "encouragement" by the framework to re-structure my equations? Maybe the implication is that most PDEs should be able to be expressed with gradients and laplacians?

[david-zwicker]

I see. It's certainly possible to use these 1d derivatives in python code, although the interface is far from ideal (I added issue #368 to improve on this in the future). In the meantime, you can use the following code to create a function derivative_x, which calculates the first derivative of an array:

grid = pde.UnitGrid([32, 32])  # define a grid

# create the function (needs to be done only once)
from pde.grids.operators.cartesian import _make_derivative, _make_derivative2
derivative_x = _make_derivative(grid, axis=0)

# define fields
source = pde.ScalarField.random_normal(grid)
result = pde.ScalarField(grid)

source.set_ghost_cells({"value": 1})  # set boundary conditions
derivative_x(source._data_full, result.data)  # apply operator

[adamg-cci]

got it. thanks for the help!

[david-zwicker]

I just noticed that we already support the simpler definition of the operator:

field._apply_operator("d_dx", bc="natural")

We should probably make this method public, to make it more accessible.

[adamg-cci]

is this numba-compilable?

[david-zwicker]

To get an operator that you can use in numba, you would use grid.make_operator("d_dx", bc="natural"), which returns a numba-compiled function that can be applied to numpy arrays.

[adamg-cci]

A few more question/clarifications:

• I have six unknowns. I only need one derivative operator, right?
• this approach is numba-compiled compatible, right?
• I am modeling a dirichlet (value=constant) case where the concentration of a species is input to the system at a constant rate, and is modified as it traverses the 1D grid. Is it correct to say that I should use the method="backward" configuration? Any guidance here would be greatly appreciated -- I am not seeing the values update as I would have expected across each timestep, as currently implemented. Namely, no spatial variation is occurring, all cells on the grid are changing in the exact same manner

[david-zwicker]

I unfortunately don't have the bandwidth to help you with the implementation of a concrete problem. Please ask a separate question if you have a concrete question on the py-pde package.

[adamg-cci]

No problem. Thank you for the help nonetheless, David!