Magnetic field diffusion in 2D and 3D. More...

Detailed Description

Sets the initial conditions for a magnetic field diffusion problem in 2 or 3 dimensions. This is a useful test to check the ability of the code to solve standard diffusion problems. The magnetic field has initially a Gaussian profile, and an anisotropic resistivity is possible. This problem has an analytical solution given, in 2D, by

$B_x(y,t) = \exp(-y^2/4\eta_zt)/\sqrt{t} \quad\quad\quad B_y(x,t) = \exp(-x^2/4\eta_zt)/\sqrt{t} \quad\quad\quad B_z(x,y,t) = \exp(-x^2/4\eta_yt)\exp(-y^2/4\eta_xt)/t$

and in 3D by

$\begin{array}{lcl} B_x(y,z,t) &=& \exp(-y^2/4\eta_zt)\exp(-z^2/4\eta_yt)/t \\ \noalign{\medskip} B_y(x,z,t) &=& \exp(-x^2/4\eta_zt)\exp(-z^2/4\eta_xt)/t \\ \noalign{\medskip} B_z(x,y,t) &=& \exp(-x^2/4\eta_yt)\exp(-y^2/4\eta_xt)/t \end{array}$

The initial condition is simply set using the previous profiles with t=1. In order to solve only the parabolic term in the induction equation ( $\nabla\times\vec{J}=\nabla\times(\eta\nabla\times\vec{B})$ ) we give to the fluid a very large intertia using a high density value. Moreover, to avoid any fluid motion, the velocity is reset to zero at each time step by using the INTERNAL_BOUNDARY.

The runtime parameters that are read from pluto.ini are

g_inputParam[ETAX]: sets the resistivity along the direction;
g_inputParam[ETAY]: sets the resistivity along the direction;
g_inputParam[ETAZ]: sets the resistivity along the direction;

Initial and final profiles of the numerical (points) and analytical (lines) solutions for a component of the magnetic field.

The configurations use both EXPLICIT and STS time integrators and different geometries are explored:

Conf.	GEOMETRY	DIM	T.STEPPING	divB	RESISTIVITY
#01	CARTESIAN	3	RK2	8W	EXPLICIT
#02	CARTESIAN	3	HANCOCK	GLM	STS
#03	CARTESIAN	3	RK3	CT	EXPLICIT
#04	CARTESIAN	3	HANCOCK	GLM	EXPLICIT
#05	POLAR	3	RK2	8W	EXPLICIT
#06	POLAR	3	RK2	8W	STS
#07	SPHERICAL	3	RK2	8W	EXPLICIT
#08	SPHERICAL	3	RK2	8W	STS
#09	SPHERICAL	3	RK3	CT	EXPLICIT
#10	CARTESIAN	2	HANCOCK	CT	EXPLICIT
#11	CARTESIAN	3	HANCOCK	CT	EXPLICIT
#12	CARTESIAN	2	HANCOCK	CT	STS
#13	SPHERICAL	3	RK2	CT	STS

Authors: A. Mignone (migno.nosp@m.ne@p.nosp@m.h.uni.nosp@m.to.i.nosp@m.t)
T. Matsakos (titos.nosp@m.@odd.nosp@m.job.u.nosp@m.chic.nosp@m.ago.e.nosp@m.du)

Date: Sept 17, 2014