# Modified Gresho Test

## January 30, 2017

To test the 2D static potential in Cholla, I’ve created a modified version of the Gresho vortex test that balances the centrifugal force created by the angular velocity profile with an artificially added centripetal acceleration. The centripetal acceleration is added in the same operator-split manner as the gravitational acceleration would be, so it is effectively a test of the 2D static potential in Cholla. This acceleration takes the form

$$g_{x} = -\mathrm{cos}(\phi)\frac{v_{\phi}^2}{r} \quad \mathrm{and} \quad g_{y} = -\mathrm{sin}(\phi)\frac{v_{\phi}^2}{r},$$

where $\phi$ is again the angle calculated according to $\phi= \mathrm{arctan}(\frac{y}{x})$ and $r$ is the distance from the center of the vortex. The acceleration is applied to the momentum and total Energy equations in an operator-split manner at the end of the hydro update, according to

where $n^{*}$ refers to values that have undergone the hydro update, but not the gravitational source term update. There is no correction to the density.

Results

I’m running the problem until t = 3.0 on a 40x40 grid with periodic boundaries, using PPMC, the exact Riemann solver, and CTU. The domain is centered at (x, y) = (0, 0). The movies below show the evolution in azimuthal velocity, pressure, and voriticty. Unlike the previous versions of the test, the pressure is now set to be a constant function of radius.

The pressure looks a little crazy, but note from the top movie that the overall order of the pressure perturbations is not large.

The L1 density errors at $t = 3.0$ are now 0.3% and 0.07% on a $20\times20$ and $40\times40$ grid, respectively. The magnitude of the errors is thus double what it was for the standard (static) Gresho test, which doesn’t seem too bad given the operator-split nature of the acceleration correction. The overall magnitude by which the azimuthal velocity goes down is approximately the same.