Robin Boundary Condition

This boundary condition has not yet been implemented in individual coordinate directions. The Robin boundary condition is:

$$f(r) = f_0 + \frac{k}{r^n}$$ (part14)

with $k$ a constant, $n$ the decay rate and $f_0$ the value at infinity. This implies:

$$\frac{\partial f}{\partial r} = - n \frac{k}{r^{n+1}}$$ (part15)

or

$$\frac{\partial f}{\partial r} = - n \frac{(f-f_0)}{r}$$ (part16)

Considering now a given Cartesian direction $x$ we get:

$$\frac{\partial f}{\partial x} = \frac{\partial f}{\partial r... ...tial r}{\partial x} = \frac{x}{r}\frac{\partial f}{\partial r}$$ (part17)

which implies

$$\frac{\partial f}{\partial x} = - n (f-f_0)\frac{x}{r^2}$$ (part18)

The equations are then finite differenced around the grid point $i+1/2$:

$$f_{i+1} - f_i = - n \Delta x \left( \frac{1}{2}(f_{i+1}+f_i) - f_0\right) \frac{x_{i+1/2}}{r^2_{i+1/2}}$$ (part19)

or

$$f_{i+1}-f_i = -n \Delta x ( (f_{i+1}+f_i)-2 f_0)\frac{x_{i+1}+x_i}{(r_{i+1}+r_i)^2}$$ (part110)

And this is then solved either for $f_i$ or $f_{i+1}$ depending on which side are we looking at.

The Robin boundary condition is registered under the name ``Robin''.