1. Introduction

Numerical modeling of magnetohydrodynamics (MHD) is an important and challenging problem addressed in numerous publications (e.g., see [1, 2]). This problem is further complicated in case of multi-flux models that account for the relative motion and interaction of particles of different nature (electrons, various species of ions, neutral atoms, and molecules) both with each other and with an external magnetic field.

This class of problems is generally solved using the fractional-step method, when complex operators are represented as a product of operators having a simpler structure. Thus, within the splitting method, the calculation of one-time step consists of a series of simpler procedures. It is obvious that difference schemes for each splitting stage should, where possible, preserve the properties of corresponding difference equations.

Note that the task of constructing reference solutions accounting for the whole range of physical processes is challenging (and often unfeasible). Existing benchmarks enable accuracy assessment of individual splitting stages rather than the simulation as a whole.

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Magnetohydrodynamic problems are naturally divided into two groups: problems for an ideal infinitely conducting plasma and problems with dissipative processes in the form of heat conduction and magnetic viscosity.

dHx dt <sup>¼</sup> <sup>ν</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>ν</sup><sup>2</sup> <sup>þ</sup> <sup>β</sup><sup>2</sup>

> ξ 0

> > 2

ffiffiffiffiffiffi γ�ν 2γ q

¼

solution Hxð Þ¼ <sup>t</sup>; <sup>z</sup> <sup>¼</sup> <sup>L</sup> <sup>1</sup> <sup>þ</sup> ð Þ <sup>C</sup>1<sup>Φ</sup> <sup>þ</sup> <sup>C</sup>2<sup>Ψ</sup> <sup>L</sup>ffiffiffiffiffi

boundary conditions can be found from equations

<sup>C</sup><sup>1</sup> <sup>¼</sup> �Hx0Φð Þ� <sup>∞</sup> Hy0Ψð Þ <sup>∞</sup>

<sup>C</sup><sup>2</sup> <sup>¼</sup> �Hx0Ψð Þþ <sup>∞</sup> Hy0Φð Þ <sup>∞</sup>

Figure 1. Profiles of field components at time t = 0.01: (a) Hx and (b) Hy.

q

where <sup>Φ</sup>ð Þ¼ <sup>ξ</sup> <sup>Ð</sup>

Since <sup>Φ</sup>ð Þ¼ <sup>∞</sup> <sup>Γ</sup> <sup>1</sup>

Let γ ¼

like

∂2 Hx <sup>∂</sup>z<sup>2</sup> <sup>þ</sup> <sup>β</sup>

given boundary and initial conditions:

∂2 Hy <sup>∂</sup>z<sup>2</sup> , dHy

dt <sup>¼</sup> <sup>ν</sup>

∂2 Hy <sup>∂</sup>z<sup>2</sup> � <sup>β</sup>

Consider the problem of a diffusion wave propagating in an unbounded medium with the

Hx ¼ Hx<sup>0</sup> þ C1Φð Þþ ξ C2Ψð Þ ξ , Hy ¼ Hy<sup>0</sup> þ C1Ψð Þ� ξ C2Φð Þ ξ , Hz ¼ Hz<sup>0</sup>

2

ffiffiffiffiffiffi γþν 2γ q

> ffiffiffi <sup>π</sup> <sup>p</sup> Hx<sup>0</sup>

ffiffiffi <sup>π</sup> <sup>p</sup> Hx<sup>0</sup>

Simulation setup: the initial data is described by Eq. (3). A bounded computational domain 0 < z < L, L = 1 is considered. For this reason, the magnetic field value taken from the analytical

¼

exp �νx<sup>2</sup>=<sup>γ</sup> � � sin <sup>β</sup>x<sup>2</sup>=<sup>γ</sup> � �dx, <sup>Ψ</sup>ð Þ¼ <sup>ξ</sup> <sup>Ð</sup>

, <sup>Ψ</sup>ð Þ¼ <sup>∞</sup> <sup>Γ</sup> <sup>1</sup>

<sup>Φ</sup>ð Þþ <sup>∞</sup> <sup>Ψ</sup>ð Þ <sup>∞</sup> ¼ � <sup>2</sup>

<sup>Φ</sup>ð Þþ <sup>∞</sup> <sup>Ψ</sup>ð Þ <sup>∞</sup> ¼ � <sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffi π γð Þ �ν 2γ q

Hð Þ¼ t; z ¼ 0 H1, Hð Þ¼ t; z ! ∞ H0, Hð Þ¼ t ¼ 0; z H0, H<sup>0</sup> ¼ ð Þ 0; 0; Hz<sup>0</sup> , H<sup>1</sup> ¼ Hx0; Hy0; Hz<sup>0</sup>

∂2 Hx <sup>∂</sup>z<sup>2</sup> , dHz

. A general solution to Eq. (2) for the self-similar variable <sup>ξ</sup> <sup>¼</sup> <sup>z</sup><sup>=</sup> ffiffiffiffiffiffiffi

ξ 0

> ffiffiffiffiffiffiffiffiffiffiffi π γð Þ þν 2γ q

> > ffiffiffiffiffiffiffiffiffiffiffi γ � ν 2γ

ffiffiffiffiffiffiffiffiffiffiffi γ þ ν 2γ

<sup>4</sup>γ<sup>t</sup> <sup>p</sup> , Hyð Þ¼ <sup>t</sup>; <sup>z</sup> <sup>¼</sup> <sup>L</sup> <sup>1</sup> <sup>þ</sup> ð Þ <sup>C</sup>1<sup>Ψ</sup> � <sup>C</sup>2<sup>Φ</sup> <sup>L</sup>ffiffiffiffiffi

r

s

exp �νx<sup>2</sup>=<sup>γ</sup> � � cos <sup>β</sup>x<sup>2</sup>=<sup>γ</sup> � �dx:

þ Hy<sup>0</sup>

� Hy<sup>0</sup>

! s

! r

dt <sup>¼</sup> <sup>0</sup>, <sup>β</sup> <sup>¼</sup> bHz0: (2)

Benchmarks for Non-Ideal Magnetohydrodynamics http://dx.doi.org/10.5772/intechopen.75713

constants C1, C<sup>2</sup> with regard to

,

:

<sup>4</sup>γ<sup>t</sup> <sup>p</sup> , Hzðt; <sup>z</sup> <sup>¼</sup>

ffiffiffiffiffiffiffiffiffiffiffi γ þ ν 2γ

ffiffiffiffiffiffiffiffiffiffiffi γ � ν 2γ

� �: (3)

4γt p looks

219

Numerous publications on the construction of difference methods for ideal magnetohydrodynamics use a standard set of test problems. These include propagation of one-dimensional Alfven waves at various angles to grid lines [3–5], Riemann problem for MHD equations [6–9], and various two-dimensional problems accounting for the presence of a uniform magnetic field [3, 5, 10]. In [11], a number of additional ideal MHD benchmarks are presented, which are basically shock-wave problems. A special class of tests includes problems with a weak magnetic field not affecting the medium motion. If there is an exact solution for a given hydrodynamic problem, the magnetic field "freezing-in" principle allows finding components of the field H Hx; Hy; Hz at any time with the known medium displacements X <sup>¼</sup> X Xð Þ <sup>0</sup>; <sup>t</sup> .

The representation in publications of the problem of testing the dissipative stage of MHD equations is much the worse. Possibly, this is owing to complexity problems that require accounting the interaction of the shock-wave processes, heat conduction, diffusion of magnetic field, and Joule heating.

Numerical simulations of some of the tests presented here have been done using the Lagrangian-Eulerian code EGIDA developed at VNIIEF [12, 13] for multi-material compressible flow simulations.

The magnetohydrodynamic equation system in one-temperature approximation modified by the Hall effect can be written in the following conservative form [2]:

$$\begin{aligned} \frac{\partial \rho}{\partial t} + div\rho \mathbf{u} &= 0, \quad \frac{\partial \rho \mathbf{u}}{dt} + \text{div}(\rho \mathbf{u} \otimes \mathbf{u} + (P + P\_H)I - 0.5 \mathbf{H} \otimes \mathbf{H}) = 0, \quad P\_H = 0.5|\mathbf{H}|^2, \\\frac{\partial H}{\partial t} + \text{div}(\mathbf{u} \otimes \mathbf{H} - \mathbf{H} \otimes \mathbf{u}) &= -rot(\nu \cdot rot \mathbf{H} + b[\mathbf{H} \times rot \mathbf{H}]), \quad \frac{\partial \rho n\_\varepsilon}{\partial t} + \text{div}(\rho n\_\varepsilon \mathbf{u}) = 0, \\\frac{\partial \Xi}{\partial t} + \text{div}((\Xi + P + P\_H)\mathbf{u} - \mathbf{H}(\mathbf{u} \cdot \mathbf{H}) - \kappa gr dT) &= 0, \quad \Xi = \rho \left(\varepsilon + 0.5|\mathbf{u}|^2\right) + P\_{H\prime} \\P = P(\rho, T), \quad \varepsilon = \varepsilon(\rho, T). \end{aligned} \tag{1}$$

where <sup>ν</sup> <sup>¼</sup> <sup>c</sup><sup>2</sup>=ð Þ <sup>4</sup>πσ is the magnetic viscosity coefficient, <sup>κ</sup> is the heat conduction factor, b ¼ c=ð Þ 4πene is a local exchange (Hall) parameter [2], and e and ne are charge and density of electrons. When writing Eq. (18), we assume that bias currents and electron inertia are negligibly small [2]. Equation system (1) differs from equation system for ideal MHD owing to diffusion terms present in the equations of energy and inductance of magnetic field.
