1D Gauge wave tests

The next Figure shows results of the error in $g_{xx}$ for 1D gauge waves with amplitude $A=.5$ using the $\hat W$-algorithm with no additional constraint adjustment or dissipation.

The 1D wave runs were stopped at $t=1000$ (1000 crossing times). At $t=1000$ the error for the coarsest grid was unacceptably large (over 100%) but there was only $\sim 20$ % error in the $\ell_\infty$ norm for the 200 point grid. For a function $\Phi_\rho$ evolved on a grid with spacing $1/(50\rho)$ with known analytic values $\Phi_{ana}$, the convergence rate ${\tilde c_r}$ is given by

$${\tilde c_r} = log_2 \big ( \frac{\vert\vert\Phi_\rho-\Phi... ...ert\vert}{\vert\vert\Phi_{2\rho}-\Phi_{ana}\vert\vert} \big ),$$ (8)

in terms of an appropriate norm $\vert\vert\, \, \vert\vert$. Using the $\ell_\infty$ norm and the $\rho=2$ and $\rho=4$ runs, we find the following convergence rate for the error in $g_{xx}$, measured at t=50 and t=500:

$${\tilde c_r^{(50)}} = 2.019, {\tilde c_r^{(500)}} = 1.677$$ (9)