diff --git a/doc/sphinx/inputNamelists/controlNamelist.rst b/doc/sphinx/inputNamelists/controlNamelist.rst index ea328431..25e096fb 100644 --- a/doc/sphinx/inputNamelists/controlNamelist.rst +++ b/doc/sphinx/inputNamelists/controlNamelist.rst @@ -242,6 +242,8 @@ domain. +-----------------------+-----------------------------------------------------------------------------------------------------------+ | map_function='ARCSIN' | Use an arcsin mapping (see `Kosloff and Tal-Ezer 1993 `_) | +-----------------------+-----------------------------------------------------------------------------------------------------------+ + | map_function='TT' | Use the mapping by `Tee and Trefethen 2006 `_ | + +-----------------------+-----------------------------------------------------------------------------------------------------------+ | map_function='JAFARI' | Use the mapping by `Jafari-Varzaneh and Hosseini 2014 `_ | +-----------------------+-----------------------------------------------------------------------------------------------------------+ @@ -282,6 +284,19 @@ regular grid. .. +If the Tee and Trefethen sinh mapping is employed, the grid points are redistributed in the following manner + +.. math:: + r=\frac{1}{2}\left(\alpha_2+\frac{\textrm{sinh}\left[A(x_{cheb}-1)+B\right]}{\alpha_1}\right) + \frac{r_i+r_o}{2} \textrm{ ,} + +where + +.. math:: + A=\frac{1}{2}\left[\textrm{sinh}(\alpha_1(1-\alpha_2))+\textrm{sinh}(\alpha_1(1+\alpha_2)) \right], \quad B = \textrm{sinh}(\alpha_1(1-\alpha_2)) + +With this mapping, :math:`\alpha_1` is directly related to the stiffness of the transition. + + If the Jafari-Varzaneh and Hosseini mapping is employed, similarly to the tangent mapping, :math:`\alpha_1` determines the degree of concentration of the grid points around :math:`x_{cheb}\!=\!\alpha_2`. This is expected to do a better job than the tangent mapping, both in terms of matrix conditioning and in terms of reducing the Gibbs phenomenon around a steep change (Allen-Cahn type of equations involved in the phase field model comes to mind).