# Finite-difference schemes for anisotropic diffusion

## Abstract

In fusion plasmas diffusion tensors are extremely anisotropic due to the high temperature and large magnetic field strength. This causes diffusion, heat conduction, and viscous momentum loss, to effectively be aligned with the magnetic field lines. This alignment leads to different values for the respective diffusive coefficients in the magnetic field direction and in the perpendicular direction, to the extent that heat diffusion coefficients can be up to 10{sup 12} times larger in the parallel direction than in the perpendicular direction. This anisotropy puts stringent requirements on the numerical methods used to approximate the MHD-equations since any misalignment of the grid may cause the perpendicular diffusion to be polluted by the numerical error in approximating the parallel diffusion. Currently the common approach is to apply magnetic field-aligned coordinates, an approach that automatically takes care of the directionality of the diffusive coefficients. This approach runs into problems at x-points and at points where there is magnetic re-connection, since this causes local non-alignment. It is therefore useful to consider numerical schemes that are tolerant to the misalignment of the grid with the magnetic field lines, both to improve existing methods and to help open the possibility of applying regular non-aligned grids. Tomore »

- Authors:

- Centrum Wiskunde and Informatica, P.O. Box 94079, 1090GB Amsterdam (Netherlands)
- (Netherlands)
- Eindhoven University of Technology (Netherlands)
- FOM Institute DIFFER, Dutch Institute for Fundamental Energy Research, Association EURATOM-FOM (Netherlands)

- Publication Date:

- OSTI Identifier:
- 22314900

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 272; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ANISOTROPY; APPROXIMATIONS; COORDINATES; DIFFUSION; DIFFUSION EQUATIONS; MAGNETIC FIELDS; MAGNETOHYDRODYNAMICS; MATHEMATICAL SOLUTIONS; PLASMA; TENSORS; THERMAL CONDUCTION

### Citation Formats

```
Es, Bram van, E-mail: es@cwi.nl, FOM Institute DIFFER, Dutch Institute for Fundamental Energy Research, Association EURATOM-FOM, Koren, Barry, and Blank, Hugo J. de.
```*Finite-difference schemes for anisotropic diffusion*. United States: N. p., 2014.
Web. doi:10.1016/J.JCP.2014.04.046.

```
Es, Bram van, E-mail: es@cwi.nl, FOM Institute DIFFER, Dutch Institute for Fundamental Energy Research, Association EURATOM-FOM, Koren, Barry, & Blank, Hugo J. de.
```*Finite-difference schemes for anisotropic diffusion*. United States. doi:10.1016/J.JCP.2014.04.046.

```
Es, Bram van, E-mail: es@cwi.nl, FOM Institute DIFFER, Dutch Institute for Fundamental Energy Research, Association EURATOM-FOM, Koren, Barry, and Blank, Hugo J. de. Mon .
"Finite-difference schemes for anisotropic diffusion". United States.
doi:10.1016/J.JCP.2014.04.046.
```

```
@article{osti_22314900,
```

title = {Finite-difference schemes for anisotropic diffusion},

author = {Es, Bram van, E-mail: es@cwi.nl and FOM Institute DIFFER, Dutch Institute for Fundamental Energy Research, Association EURATOM-FOM and Koren, Barry and Blank, Hugo J. de},

abstractNote = {In fusion plasmas diffusion tensors are extremely anisotropic due to the high temperature and large magnetic field strength. This causes diffusion, heat conduction, and viscous momentum loss, to effectively be aligned with the magnetic field lines. This alignment leads to different values for the respective diffusive coefficients in the magnetic field direction and in the perpendicular direction, to the extent that heat diffusion coefficients can be up to 10{sup 12} times larger in the parallel direction than in the perpendicular direction. This anisotropy puts stringent requirements on the numerical methods used to approximate the MHD-equations since any misalignment of the grid may cause the perpendicular diffusion to be polluted by the numerical error in approximating the parallel diffusion. Currently the common approach is to apply magnetic field-aligned coordinates, an approach that automatically takes care of the directionality of the diffusive coefficients. This approach runs into problems at x-points and at points where there is magnetic re-connection, since this causes local non-alignment. It is therefore useful to consider numerical schemes that are tolerant to the misalignment of the grid with the magnetic field lines, both to improve existing methods and to help open the possibility of applying regular non-aligned grids. To investigate this, in this paper several discretization schemes are developed and applied to the anisotropic heat diffusion equation on a non-aligned grid.},

doi = {10.1016/J.JCP.2014.04.046},

journal = {Journal of Computational Physics},

number = ,

volume = 272,

place = {United States},

year = {Mon Sep 01 00:00:00 EDT 2014},

month = {Mon Sep 01 00:00:00 EDT 2014}

}