Analysis of the Numerical Diffusion in Anisotropic Mediums: Benchmarks for Magnetic Field Aligned Meshes in Space Propulsion Simulations

Abstract

This manuscript explores numerical errors in highly anisotropic diffusion problems. First, the paper addresses the use of regular structured meshes in numerical solutions versus meshes aligned with the preferential directions of the problem. Numerical diffusion in structured meshes is quantified by solving the classical anisotropic diffusion problem; the analysis is exemplified with the application to a numerical model of conducting fluids under magnetic confinement, where rates of transport in directions parallel and perpendicular to a magnetic field are quite different. Numerical diffusion errors in this problem promote the use of magnetic field aligned meshes (MFAM). The generation of this type of meshes presents some challenges; several meshing strategies are implemented and analyzed in order to provide insight into achieving acceptable mesh regularity. Second, Gradient Reconstruction methods for magnetically aligned meshes are addressed and numerical errors are compared for the structured and magnetically aligned meshes. It is concluded that using the latter provides a more correct and straightforward approach to solving problems where anisotropicity is present, especially, if the anisotropicity level is high or difficult to quantify. The conclusions of the study may be extrapolated to the study of anisotropic flows different from conducting fluids.

Numerical treatment of a magnetized electron fluid model within an electromagnetic plasma thruster simulation code

“…Hall2De implements a fully 2D treatment of the electron fluid equations on a magnetic field aligned mesh (MFAM or magnetic mesh for short) and is applicable to any axisymmetric magnetic topology. At large values of the Hall parameter, the use of a magnetic mesh is almost mandatory to avoid the strong numerical diffusion between the directions parallel and perpendicular to the magnetic field [27,28] (incidentally, in order to prevent the PIC-related statistical noise, Hall2De opted also for a fluid treatment of heavy species).…”

“…The goal of this paper is to analyze the numerical algorithms used to solve the electron continuity and momentum equations of the electron fluid on a given MFAM. The strategies for the generation of a suitable 2D magnetic mesh, based on geometric quality indicators, were already discussed in [28,33]. The challenges here to obtain accurate numerical algorithms come from several sides: (1) the high irregularity of the cells; (2) the large anisotropy in electron conductivity caused by the magnetic field; (3) the treatment of non-magnetically aligned boundary cells, and (4) the presence of null singular points.…”

Using electron fluid models to analyze plasma thruster discharges

“…The big issue with MFAMs, well illustrated in Fig. 1(a), is they have very irregular cells and the mesh boundaries are not magnetically aligned [39]. This made delicate to derive accurate algorithms for spatial (36) derivatives in MFAMs, mainly at boundary cells [40,41]; for instance, algorithms based on weighted-least-squares methods lead to numerical diffusion and large errors in Eq.…”

Numerical study of viscosity and heat flux role in heavy species dynamics in Hall thruster discharge

“…The rectangular mesh is a basic mesh, which covers the entire simulation region uniformly. The MFA mesh is constructed above the regular one using the exponential compression technique [49,50]. The triangular mesh is constructed in such a way that the vertices of triangular cells are placed at the centers of MFA cells.…”