## 2 Numerical Diffusion in Regular, Structured, Meshes: The Anisotropic Diffusion Problem
### Problem Formulation  
The mathematical model describing the flow motion as derived by [PERSON] (1958), considering plasma quasineutrality \(n_{e}\approx n_{i}=n\), for an infinite 2D slab with conducting walls under a straight magnetic field can be expressed as  
\[\frac{dn}{dt}=\left[ \begin{array}{cc}\frac{\mu_{el}|D_{\perp}-\mu_{\parallel} D_{\perp}}{\mu_{el}|^{-}\mu_{\parallel}}&0\\ 0&\frac{\mu_{el}|D_{\parallel}-\mu_{\parallel}D_{el}|}{\mu_{el}|^{-}\mu_{ \parallel}}\end{array} \right]\left[ \begin{array}{c}\frac{\partial^{2}n}{ \partial\gamma^{2}}\\ \frac{\partial^{2}n}{\partial\gamma^{2}}\end{array} \right], \tag{1}\]  
where \(n\) is the fluid density, the \(\mu\) terms are the mobility coefficients for both species, and \(D_{\perp}\) and \(D_{\parallel}\) are, respectively, the diffusion coefficients across the magnetic field and the \"free\" or natural diffusion coefficient. The former coefficient accounts for classical collision mechanisms, while the latter one appears in the non-confining direction parallel to the magnetic field.The variables \(x^{\prime}\) and \(y^{\prime}\) are, respectively, the coordinates associated to the directions perpendicular (\(\vec{1}_{\perp}\)) and parallel (\(\vec{1}_{\parallel}\)) to the magnetic field as shown in Figure 1.  
Figure 1: (**a**) The marked region represents a typical axisymmetric simulation domain for a Hall Effect Thruster geometry; (**b**) Magnetic Field Aligned Mesh for the SPT-100 Hall Effect Thruster magnetic topology; (**c**) Detail of a mesh element. For the sake of simplicity, the effects of the ambipolar perpendicular electric field for non-conducting walls can be neglected and the diffusion term can be considered uniform across the flow domain. Under these assumptions, the diffusion terms can be written as:  
\[\mathcal{D}^{{}^{\prime}}=\left[ \begin{array}{cc}D_{\perp}&0\\ 0&D_{\parallel}\end{array} \right]. \tag{2}\]  
Then, the diffusion coefficient ratio can be defined as \(\Theta=\frac{D_{\parallel}}{D_{\perp}}\) and it is the main parameter in this study, as it determines the rate of numerical diffusion. Formally, the flow behavior can be modeled as the classical diffusion equation (Equation (1)), taking into account the flux vector \(\Gamma\) and strictly positive diffusion coefficients as  
\[\frac{dn}{dt}=\
abla^{{}^{\prime}}\times\left[\mathcal{D}^{{}^{\prime}}\times \
abla^{{}^{\prime}}n\right]=\
abla^{{}^{\prime}}\times\vec{\Gamma}=\
abla^{{} ^{\prime}}\times\left[ \begin{array}{c}\Gamma_{x^{\prime}}\\ \Gamma_{y}^{\prime}\end{array} \right] \tag{3}\]  
The numerical diffusion problem arises when the diffusion equation (Equation (3)) is discretized in a coordinate system which is not aligned with the perpendicular and parallel directions of the magnetic field, such as a Cartesian coordinate system for a randomly aligned magnetic field as represented in Figure 2. For the sake of simplicity, let's consider that magnetic field lines are straight (but the problem is easily generalizable to a curved field) and magnetic versors (\(\vec{1}_{\perp}\), \(\vec{1}_{\parallel}\) defined in Figure 1) are rotated a given angle \(\alpha\) with respect to the Cartesian system as shown in Figure 2.  
The rotation matrix \(\mathcal{R}\) allows for transforming Equation (3) into the Cartesian coordinate frame by using the relations:  
\[\mathcal{R} = \left[ \begin{array}{cc}\cos\alpha&-\sin\alpha\\ \sin\alpha&\cos\alpha\end{array} \right]; \tag{4}\] \[\vec{\Gamma} = \mathcal{R}\times\vec{\Gamma};\] \[\
abla^{{}^{\prime}}n = \mathcal{R}\times\
abla n.\]  
Replacing in Equation (3), results in the diffusion equation in Cartesian coordinates which satisfies:  
\[\frac{dn}{dt}=\
abla\times\left[\mathcal{R}^{-1}\mathcal{D}^{{}^{\prime}} \mathcal{R}\times\
abla n\right]; \tag{5}\]  
with the new diffusion matrix \(\mathcal{D}\) being:  
\[\mathcal{D}=\mathcal{R}^{-1}\mathcal{D}^{{}^{\prime}}\mathcal{R}=\left[