Radiation Boundary Condition

This is a two level scheme. Grid functions are given for the current time level (to which the BC is applied) as well as grid functions from a past timelevel which are needed for constructing the boundary condition. The grid function of the past time level needs to have the same geometry. Currently radiative boundary conditions can only be applied with a stencil width of one in each direction.

The radiative boundary condition that is implemented is

$$f = f_0 + \frac{u(r-vt)}{r}+\frac{h(r+vt)}{r}$$ (part11)

That is, outgoing radial waves with a $1/r$ fall off, and the correct asymptotic value $f_0$ are assumed, including the possibility of incoming waves (these incoming waves should be modeled somehow).

Condition A1.1 above leads to the differential equation:

$$\frac{x^i}{r}\frac{\partial f}{\partial t} + v \frac{\partia... ...partial x^i} +\frac{v x^i}{r^2} (f-f_0) = H \frac{v x^i}{r^2}$$ (part12)

where $x^i$ is the normal direction to the given boundaries, and $H = 2 dh(s)/ds$.

At a given boundary only the derivatives in the normal direction are considered. Notice that $u(r-vt)$ has disappeared, but we still do not know the value of $H$.

To get $H$ we do the following: The expression is evaluated one point in from the boundary and solved for $H$ there. Now we need a way of extrapolating $H$ to the boundary. For this, assume that $H$ falls off as a power law:

$$H = \frac{k}{r^n} \qquad \mbox{which gives} \qquad d_i H = - n \frac{H}{r}$$ (part13)

The value of $n$ is defined by the parameter radpower. If this parameter is negative, $H$ is forced to be zero (this corresponds to pure outgoing waves and is the default).

The observed behavior is the following: Using $H=0$ is very stable, but has a very bad initial transient. Taking $n$ to be 0 or positive improves the initial behavior considerably, but introduces a drift that can kill an evolution at very late times. Empirically, the best value found so far is $n=2$, for which the initial behavior is very nice, and the late time drift is quite small.

Another problem with this condition is that it does not use the physical characteristic speed, but rather it assumes a wave speed of $v$, so the boundaries should be out in the region where the characteristic speed is constant. Notice that this speed does not have to be 1.

The radiation boundary condition is registered under the name ``Radiation''.