Problem 7:

Again use the symmetric 9-point molecule discussed in class and in "The Fully Averaged Implicit Scheme" by M. Alcubierre.

As a most simple example of a non-uniform shift, consider a shift that is a linear function of position in the moving coordinate frame:

beta(x) = ax, where a is a constant parameter.

Then the metric is:

g11 = exp(at),

and the coefficient of -dQ/dx is given by:

uc_gamma= a^2 x.

Use radiation boundary conditions, and consider right and left travelling waves, for both positive and negatve a. The constant ashould be chosen so that beta > 1 (or beta < -1) over about half the computational grid.