## 2 Numerical Diffusion in Regular, Structured, Meshes: The Anisotropic Diffusion Problem
### Problem Formulation
\begin{array}{cc}D_{\perp}\cos^{2}\alpha+D_{\parallel}\sin^{2}\alpha&\left( D_{\parallel}-D_{\perp}\right)\cos\alpha\sin\alpha\\ \left(D_{\parallel}-D_{\perp}\right)\cos\alpha\sin\alpha&D_{\parallel}\cos^{2 }\alpha+D_{\perp}\sin^{2}\alpha\end{array} \right].\]  
Figure 2: Regular structured mesh with tilted Magnetic field. Blue lines are aligned with the magnetic field \(\vec{B}\). The variables \(x\) and \(y\) are the Cartesian coordinates (reference) and \(x^{\prime}\) and \(y^{\prime}\) are the coordinates aligned with the magnetic field. where the \(n_{i}\) terms are the ones illustrated in Figure 4. Equation (8) is applied to each volume element of the domain leading to a system of algebraic equations, which can be written as: \([A]\left\{n\right\}=\left\{B\right\}\). Where \(\left\{n\right\}\) are the unknowns of the problem and \([A]\) is a sparse matrix. The system is solved numerically to obtain the temporal evolution of the anisotropic diffusion problem (for a given value of \(\Theta=\
icefrac{{D_{\parallel}}}{{D_{\perp}}}\)) for the limit case of non-permeable walls under a magnetic field forming an angle \(\alpha\) with the Cartesian system. The system evolution for \(\alpha=-45^{\circ}\) and \(\Theta\rightarrow\infty\) is presented in Figure 5, illustrating the appearance of numerical diffusion, since the _red line_ represents a magnetic field line and as such should be impermeable to the plasma. The errors due to numerical diffusion could be quantified computing the average error density for the elements in the low density zone (at the left side of the red line in Figure 5). The ideal behavior will be to consider that the plasma should be perfectly confined, as it is set in the initial condition, that is:  
\[\left\langle error(t)\right\rangle=\frac{1}{N_{elem}}\sum_{i=1}^{N}\frac{|n_{i }(t)-n(t=0)|}{n(t=0)}. \tag{9}\]  
Figure 6 shows the evolution of the error due to numerical diffusion for the \(\Theta\rightarrow\infty\) case, with different values of \(\alpha\), and different number of mesh elements. Figure 6a shows that the error increases with time and Figure 6b shows how this error is reduced when increasing the number of cells in the computational domain. Another interesting analysis corresponds to the concept of effective diffusion coefficient ratio, \(\Theta_{eff}\). This coefficient provides an equivalent diffusive mass flux in a problem in which the mesh and magnetic fields are aligned, for a given simulation domain, mesh resolution, initial condition, and the parameters \(\alpha\) and \(\Theta\). Figure 4: Schematic of central differences in a cartesian mesh. The central node on each volume are indicated, \(n_{i}\), for the six cells considered. The derivative at the faces are computed using the nearest nodes. Figure 5: Time evolution for a simple diffusion problem on a Cartesian mesh for the case \(\Theta=\
icefrac{{D_{\parallel}}}{{D_{\perp}}}\) and \(\alpha=-45^{\circ}\). Non-dimensionalized variables are used. Table 1 presents results for the effective diffusion coefficient for different parameter sets (\(\Theta\), \(\alpha\), and number of cells) for the problem described by Equation (6). The problem is solved through the explicit Euler/FVM discretization of Equation (7). Results are in agreement with those from the literature [16; 18]. Numerical diffusion leads to a reduction in the anisotropicity of the system, as accounted by the effective \(\Theta\). For the explored sets, the errors are just slightly dependent on the misalignment of the magnetic field and the mesh, \(\alpha\), when the mesh resolution is sufficiently fine. Errors are however very noticeable for lower resolution meshes and the accuracy is only partially improved through refinement for the high anisotropicity cases.