Stable finite difference schemes for the magnetic induction equation with Hall effect

Abstract

We consider a sub-model of the Hall-MHD equations: the so-called magnetic induction equations with Hall effect. These equations are non-linear and include third-order spatial and spatio-temporal mixed derivatives. We show that the energy of the solutions is bounded and design finite difference schemes that preserve the energy bounds for the continuous problem. We design both divergence preserving schemes and schemes with bounded divergence. We present a set of numerical experiments that demonstrate the robustness of the proposed schemes.

Appendix: Finite difference operators

The different operators used in our numerical experiment, are based on one dimensional operators coupled together with Kronecker product. The one dimensional operators are given for q=x,y,z in matrix form:

• Second order central difference

  $$D^{(2)}_{q}=P_q^{-1}Q= \frac{1}{2 \varDelta q} \begin{pmatrix} -2 & 2 & & & \\ -1 & 0 & 1 & &\\ &\ddots& \ddots& \ddots&\\ && -1 & 0 & 1 \\ &&& -2 & 2 \end{pmatrix} , \qquad P_q=\varDelta q \begin{pmatrix} \frac{1}{2} & & & &\\ & 1 & & & \\ &&\ddots&&\\ &&&1& \\ &&&&\frac{1}{2} \end{pmatrix} . $$

• Fourth order central difference

  

Combining this operators we obtain the two spatial discretisation used in the numerical experiments.

We give the discrete derivative for the x direction, the ones for the other spatial directions are defined analogously.

Standard second and fourth order operator are

$$\mathfrak {d}_x=D^{(k)}_x \otimes I_y \otimes I_z, \quad k=2,4 $$

where I _q are the identity matrices.