How can one enforce a left boundary condition in _make_derivative(some_grid, method="forward")?

[HannoSpreeuw]

I have been running this code, with Depths a CartesianGrid and cCa a ScalarField which uses the Depths grid

self.forward_diff = _make_derivative(Depths, method="forward")
cCa_grad_forw = cCa.copy()
cCa_grad_forw.data[:] = 0
cCa.set_ghost_cells(self.bc_cCa)
self.forward_diff(cCa._data_full, out = cCa_grad_forw.data)

When looking at cCa_grad_forw or rather on how the pde integrates using this code, it seems the left bc for cCa, from self.bc_cCa, is not imposed, probably due to the way _make_derivative with method="forward" is
implemented.
Looking at the source code:

elif method == "forward":
    out[i - 1] = (arr[i + 1] - arr[i]) / dx

It seems the 'left' side of the data, which would have to come from arr[i-1] and taken from a ghost cell, is never read when forward differencing is applied?

How to fix this?

[david-zwicker]

Yes, this is expected behavior. If you only use a single derivative, imposing a boundary conditions on one side is sufficient.

[HannoSpreeuw]

Thanks, but now I get solutions which do not respect the boundary condition.

I do, however, want to use a forward difference, to gain better accuracy.
I am trying to account of advection, in this case a flow from higher depths to lower depths, so I would like to make use of upstream values.

[HannoSpreeuw]

I could redefine all derivatives.

                if method == "central":
                    out[i - 1] = (arr[i] - arr[i - 2]) / (2 * dx)
                elif method == "forward":
                    out[i - 1] = (arr[i] - arr[i-1]) / dx
                elif method == "backward":
                    out[i - 1] = (arr[i-1] - arr[i - 2]) / dx

[david-zwicker]

I guess I understand what you're trying to do, but I don't think the implementation you're suggesting is correct. This is because the index into the out and arr fields are not directly comparable. The out array has N entries corresponding to the N support points in this 1D case. In contrast, the arr field has N+2 entries, because it accounts for the virtual points (or ghost points) outside either boundary. Consequently, the entry arr[i] co-localizes with point out[i - 1] and the original definitions are the proper forward, backward, and central differences. I agree that this can be confusing and a better comment would go a long way.