## 3 Meshing Strategies for Curvilinear Meshes Aligned with the Preferential Directions of the Problem  
The analysis presented in the previous section concludes that numerical diffusion is an important issue in numerical simulations of highly anisotropic flow, and therefore some steps should be taken to avoid it. One of the possible solutions is the use of meshes aligned with preferential directions of the problem, as defined by the cause of the anisotropy. In the case of conducting fluids, this cause is the magnetic field, and thus the use of a Magnetic Field Aligned Mesh (MFAM) is proposed. A stationary MFAM can be built when the magnetic field in the simulated domain can be considered to be both irrotational, \(\
abla\times\overrightarrow{B}\approx 0\), solenoidal, \(\
abla\times\overrightarrow{B}=0\), and stationary, \(\frac{\partial\overrightarrow{B}}{\partial t}=0\). The irrotational approximation is valid whenever the effect of the flow discharge over the external \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Mesh resolution** & \(\mathbf{\Theta_{eff}\sim...}\) & \(\mathbf{\Theta\rightarrow\infty}\) & \(\mathbf{\Theta=100}\) & \(\mathbf{\Theta=10}\) \\ \hline \multirow{3}{*}{50 elements} & \(\alpha=20^{\circ}\) & \(\sim\)5 & \(\sim\)4 & \(\sim\)3 \\  & \(\alpha=45^{\circ}\) & \(\sim\)5 & \(\sim\)5 & \(\sim\)3 \\  & \(\alpha=60^{\circ}\) & \(\sim\)5 & \(\sim\)5 & \(\sim\)3 \\ \hline \multirow{3}{*}{500 elements} & \(\alpha=20^{\circ}\) & \(\sim\)20 & \(\sim\)15 & \(\sim\)5 \\  & \(\alpha=45^{\circ}\) & \(\sim\)25 & \(\sim\)15 & \(\sim\)5 \\  & \(\alpha=60^{\circ}\) & \(\sim\)20 & \(\sim\)15 & \(\sim\)6 \\ \hline \multirow{3}{*}{5000 elements} & \(\alpha=20^{\circ}\) & \(\sim\)50 & \(\sim\)25 & \(\sim\)8 \\  & \(\alpha=45^{\circ}\) & \(\sim\)50 & \(\sim\)30 & \(\sim\)8 \\ \cline{1-1}  & \(\alpha=60^{\circ}\) & \(\sim\)50 & \(\sim\)30 & \(\sim\)7 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Effective diffusion coefficient ratio (\(\Theta_{eff}\)) for different mesh resolutions, alignment of the magnetic field (\(\alpha\)) and diffusion coefficient ratios (\(\Theta\)). Figure 6: (**a**) Error, computed as defined in Equation (9) vs. time for an infinite diffusion ratio and various inclinations of the magnetic field for a mesh resolution of 5000 elements; (**b**) Error vs. time for a diffusion ratio equal to infinite for various mesh resolutions; the inclination is fixed at 45 degrees. magnetic field is small compared to the field generated by the magnetic circuit. If these conditions are not fulfilled, then the magnetic field will be dependent on the flow motion and thus the mesh will not be constant during the simulation. In a typical axisymmetric simulation domain (Figure 1) and in 2D planar one, the above magnetic field equations allow one to derive the differential relations below for the magnetic stream-line function \(\lambda\) and the magnetic potential function \(\sigma\), which can be integrated and become the variables of a curvilinear system of magnetic coordinates. \[\
abla\times\overrightarrow{B}=0\rightarrow\
abla^{2}\lambda=0\rightarrow \frac{\partial\lambda}{\partial r}=-rB_{z};\frac{\partial\lambda}{\partial z }=rB_{r} \tag{10}\]  
\[\
abla\times\overrightarrow{B}\approx 0\rightarrow\
abla\sigma=\vec{B} \rightarrow\frac{\partial\sigma}{\partial r}=B_{r};\frac{\partial\sigma}{ \partial z}=B_{z} \tag{11}\]  
Lines of constant \(\lambda\) always follow the local direction of the magnetic field (\(\vec{1}_{\parallel}\)) and lines of constant \(\sigma\) are always in the perpendicular direction to the magnetic field (\(\vec{1}_{\perp}\)) as shown in Figure 1a. The intersections of these isolines define the points that are the vertexes of the volumes of the constructed mesh.