# A stochastic approach to uncertainty in the equations of MHD kinematics

## Abstract

The magnetohydrodynamic (MHD) kinematics model describes the electromagnetic behavior of an electrically conducting fluid when its hydrodynamic properties are assumed to be known. In particular, the MHD kinematics equations can be used to simulate the magnetic field induced by a given velocity field. While prescribing the velocity field leads to a simpler model than the fully coupled MHD system, this may introduce some epistemic uncertainty into the model. If the velocity of a physical system is not known with certainty, the magnetic field obtained from the model may not be reflective of the magnetic field seen in experiments. Additionally, uncertainty in physical parameters such as the magnetic resistivity may affect the reliability of predictions obtained from this model. By modeling the velocity and the resistivity as random variables in the MHD kinematics model, we seek to quantify the effects of uncertainty in these fields on the induced magnetic field. We develop stochastic expressions for these quantities and investigate their impact within a finite element discretization of the kinematics equations. We obtain mean and variance data through Monte Carlo simulation for several test problems. Toward this end, we develop and test an efficient block preconditioner for the linear systems arising frommore »

- Authors:

- Applied Mathematics & Statistics, and Scientific Computation Program, University of Maryland, College Park, MD (United States)
- Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD (United States)

- Publication Date:

- OSTI Identifier:
- 22465603

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 284; 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; COMPUTERIZED SIMULATION; FINITE ELEMENT METHOD; ITERATIVE METHODS; MAGNETIC FIELDS; MAGNETOHYDRODYNAMICS; MONTE CARLO METHOD; STOCHASTIC PROCESSES

### Citation Formats

```
Phillips, Edward G., E-mail: egphillips@math.umd.edu, and Elman, Howard C., E-mail: elman@cs.umd.edu.
```*A stochastic approach to uncertainty in the equations of MHD kinematics*. United States: N. p., 2015.
Web. doi:10.1016/J.JCP.2014.12.002.

```
Phillips, Edward G., E-mail: egphillips@math.umd.edu, & Elman, Howard C., E-mail: elman@cs.umd.edu.
```*A stochastic approach to uncertainty in the equations of MHD kinematics*. United States. doi:10.1016/J.JCP.2014.12.002.

```
Phillips, Edward G., E-mail: egphillips@math.umd.edu, and Elman, Howard C., E-mail: elman@cs.umd.edu. Sun .
"A stochastic approach to uncertainty in the equations of MHD kinematics". United States.
doi:10.1016/J.JCP.2014.12.002.
```

```
@article{osti_22465603,
```

title = {A stochastic approach to uncertainty in the equations of MHD kinematics},

author = {Phillips, Edward G., E-mail: egphillips@math.umd.edu and Elman, Howard C., E-mail: elman@cs.umd.edu},

abstractNote = {The magnetohydrodynamic (MHD) kinematics model describes the electromagnetic behavior of an electrically conducting fluid when its hydrodynamic properties are assumed to be known. In particular, the MHD kinematics equations can be used to simulate the magnetic field induced by a given velocity field. While prescribing the velocity field leads to a simpler model than the fully coupled MHD system, this may introduce some epistemic uncertainty into the model. If the velocity of a physical system is not known with certainty, the magnetic field obtained from the model may not be reflective of the magnetic field seen in experiments. Additionally, uncertainty in physical parameters such as the magnetic resistivity may affect the reliability of predictions obtained from this model. By modeling the velocity and the resistivity as random variables in the MHD kinematics model, we seek to quantify the effects of uncertainty in these fields on the induced magnetic field. We develop stochastic expressions for these quantities and investigate their impact within a finite element discretization of the kinematics equations. We obtain mean and variance data through Monte Carlo simulation for several test problems. Toward this end, we develop and test an efficient block preconditioner for the linear systems arising from the discretized equations.},

doi = {10.1016/J.JCP.2014.12.002},

journal = {Journal of Computational Physics},

number = ,

volume = 284,

place = {United States},

year = {Sun Mar 01 00:00:00 EST 2015},

month = {Sun Mar 01 00:00:00 EST 2015}

}