Scale‐up non‐Cartesian coordinates - cmyoo/cosmos GitHub Wiki

Scale-up non-Cartesian coordinates

In this code, we employ the non-Cartesian coordinate system $(x,y,z)$, firstly implemented in Ref. [1] (see also Sample-Flat-Test). The non-Cartesian coordinates $x^i$ are related to the Cartesian coordinates $X^i$ as follows:

$$ X^i=x^i-\frac{\eta}{1+\eta}\frac{L}{\pi}\sin\left(\frac{\pi}{L}x^i\right). $$

This functional form is compatible with the boundary conditions adopted in sample codes and satisfies $x^i=0$ at $X^i=0$ and $x^i=L$ at $X^i=L$.

At the origin, the infinitesimal interval in the Cartesian coordinate $\Delta X$ is covered by the non-Cartesian coordinate interval $\Delta x=(1+\eta)\Delta X$. Therefore, the central part is enlarged in this non-Cartesian coordinate system. The value of $\eta$ is set to $10$ for the sample code Adiabatic spherical. The value of $\eta$ can be controlled by the parameter "amp" in par_ini.d.

References

1. C.-M. Yoo, T. Ikeda, and H. Okawa, "Gravitational Collapse of a Massless Scalar Field in a Periodic Box," Class. Quant. Grav. 36, 075004 (2019), arXiv:1811.00762 [gr-qc].