1D Shifted gauge wave tests

The following Figure shows the error in $g_{xx}$ for the 1D shifted ave with A=.5, the ${\hat W}$ algorithm.

The convergence rate for the error in $g_{xx}$, measured at t=50, is:

$${\tilde c_r^{(50)}} = 2.135$$ (11)

A dramatic increased performance of the $\hat W$ algorithm results from the constraint adjustment:

$$A^{\mu\nu}= -\frac {c}{\sqrt{-g}} {\cal C}^\alpha \partial_\alpha (\sqrt{-g}g^{\mu\nu}) , \, c>0$$ (12)

As shown in the next Figure for 100 gridpoints, this adjustment with $c=1$ keeps the error in $g_{xx}$ under control for $t=1000$, as compared with $t \approx 200$ for the unadjusted run.

The constraint damping:

$$A^{\mu\nu}= -\lambda {\cal C}^{(\mu}\nabla^{\nu)}t , \, \lambda>0$$ (13)

also leads to improvement over the unadjusted case but the results are more modest. As shown in the next Figure for $\lambda=1$, the errors in $g_{xx}$ for the undamped and damped cases are roughly the same at 200.

Although the damped case runs longer it produces a highly oscillatory error which leads to unacceptable error. These results cannot be significantly improved through other choices of damping coefficient $\lambda$ or by using the normal direction $\nabla^\alpha t$ instead of the evolution direction $t^\alpha$.

The numerical dissipation also produces a modest improvement in performance, as the following Figure shows, with the dissipation coefficients set to $0.1$.

However, as we can see in the previous figure, dissipation is not effective in controlling the large oscillations in error produced by constraint damping.