# Implementation of the full viscoresistive magnetohydrodynamic equations in a nonlinear finite element code

## Abstract

Numerical simulations form an indispensable tool to understand the behavior of a hot plasma that is created inside a tokamak for providing nuclear fusion energy. Various aspects of tokamak plasmas have been successfully studied through the reduced magnetohydrodynamic (MHD) model. The need for more complete modeling through the full MHD equations is addressed here. Our computational method is presented along with measures against possible problems regarding pollution, stability, and regularity. The problem of ensuring continuity of solutions in the center of a polar grid is addressed in the context of a finite element discretization of the full MHD equations. A rigorous and generally applicable solution is proposed here. Useful analytical test cases are devised to verify the correct implementation of the momentum and induction equation, the hyperdiffusive terms, and the accuracy with which highly anisotropic diffusion can be simulated. A striking observation is that highly anisotropic diffusion can be treated with the same order of accuracy as isotropic diffusion, even on non-aligned grids, as long as these grids are generated with sufficient care. This property is shown to be associated with our use of a magnetic vector potential to describe the magnetic field. Several well-known instabilities are simulated to demonstratemore »

- Authors:

- Centrum Wiskunde & Informatica, P.O. Box 94079, 1090 GB Amsterdam (Netherlands)
- (Netherlands)
- Dutch Institute for Fundamental Energy Research, P.O. Box 6336, 5600 HH Eindhoven (Netherlands)
- ITER Organization, Route de Vinon sur Verdon, 13115 Saint Paul Lez Durance (France)
- Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven (Netherlands)

- Publication Date:

- OSTI Identifier:
- 22572322

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 316; Other Information: Copyright (c) 2016 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; BALLOONING INSTABILITY; COMPUTERIZED SIMULATION; DIFFUSION; EQUATIONS; FINITE ELEMENT METHOD; HOT PLASMA; MAGNETIC FIELDS; MAGNETOHYDRODYNAMICS; NONLINEAR PROBLEMS; TEARING INSTABILITY; THERMONUCLEAR REACTORS; TOKAMAK DEVICES

### Citation Formats

```
Haverkort, J.W., Dutch Institute for Fundamental Energy Research, P.O. Box 6336, 5600 HH Eindhoven, Blank, H.J. de, Huysmans, G.T.A., Pratt, J., and Koren, B., E-mail: b.koren@tue.nl.
```*Implementation of the full viscoresistive magnetohydrodynamic equations in a nonlinear finite element code*. United States: N. p., 2016.
Web. doi:10.1016/J.JCP.2016.04.007.

```
Haverkort, J.W., Dutch Institute for Fundamental Energy Research, P.O. Box 6336, 5600 HH Eindhoven, Blank, H.J. de, Huysmans, G.T.A., Pratt, J., & Koren, B., E-mail: b.koren@tue.nl.
```*Implementation of the full viscoresistive magnetohydrodynamic equations in a nonlinear finite element code*. United States. doi:10.1016/J.JCP.2016.04.007.

```
Haverkort, J.W., Dutch Institute for Fundamental Energy Research, P.O. Box 6336, 5600 HH Eindhoven, Blank, H.J. de, Huysmans, G.T.A., Pratt, J., and Koren, B., E-mail: b.koren@tue.nl. Fri .
"Implementation of the full viscoresistive magnetohydrodynamic equations in a nonlinear finite element code". United States.
doi:10.1016/J.JCP.2016.04.007.
```

```
@article{osti_22572322,
```

title = {Implementation of the full viscoresistive magnetohydrodynamic equations in a nonlinear finite element code},

author = {Haverkort, J.W. and Dutch Institute for Fundamental Energy Research, P.O. Box 6336, 5600 HH Eindhoven and Blank, H.J. de and Huysmans, G.T.A. and Pratt, J. and Koren, B., E-mail: b.koren@tue.nl},

abstractNote = {Numerical simulations form an indispensable tool to understand the behavior of a hot plasma that is created inside a tokamak for providing nuclear fusion energy. Various aspects of tokamak plasmas have been successfully studied through the reduced magnetohydrodynamic (MHD) model. The need for more complete modeling through the full MHD equations is addressed here. Our computational method is presented along with measures against possible problems regarding pollution, stability, and regularity. The problem of ensuring continuity of solutions in the center of a polar grid is addressed in the context of a finite element discretization of the full MHD equations. A rigorous and generally applicable solution is proposed here. Useful analytical test cases are devised to verify the correct implementation of the momentum and induction equation, the hyperdiffusive terms, and the accuracy with which highly anisotropic diffusion can be simulated. A striking observation is that highly anisotropic diffusion can be treated with the same order of accuracy as isotropic diffusion, even on non-aligned grids, as long as these grids are generated with sufficient care. This property is shown to be associated with our use of a magnetic vector potential to describe the magnetic field. Several well-known instabilities are simulated to demonstrate the capabilities of the new method. The linear growth rate of an internal kink mode and a tearing mode are benchmarked against the results of a linear MHD code. The evolution of a tearing mode and the resulting magnetic islands are simulated well into the nonlinear regime. The results are compared with predictions from the reduced MHD model. Finally, a simulation of a ballooning mode illustrates the possibility to use our method as an ideal MHD method without the need to add any physical dissipation.},

doi = {10.1016/J.JCP.2016.04.007},

journal = {Journal of Computational Physics},

number = ,

volume = 316,

place = {United States},

year = {Fri Jul 01 00:00:00 EDT 2016},

month = {Fri Jul 01 00:00:00 EDT 2016}

}