# 1D Magnetohydrodynamics

The nonlinear system of hyperbolic PDEs that describes the conservation laws for magnetohydrodynamics (MHD) is a `rich' example model to test my moving mesh method on.
Below are some example results from 1, 1.5 and 1.75D MHD simulations.

### What is 1.5 or 1.75D?

For 1.5D: We take the general 2D PDE formulation, write out all gradients, and set all

*y* partial derivatives to zero.

For 1.75D: We take the general 3D PDE formulation, write out all gradients, and set all

*y* and

*z* partial derivatives to zero.

In other words: we consider the 2- or 3D system to be in an equilibrium state in the *y* and *z* directions; the solution dynamics depend only on *x* and *t*.

### Physical staircasing in an oscillating plasma sheet

A high density, high pressure sheet is placed in a vacuum between two conducting, reflective walls, with a slight pressure imbalance between the left and right vaccuum. Apart from the slow oscillation that sets in, fast magnetosonic waves in the vaccuum areas cause the formation of a physical staircase within the plasma sheet.

**More details:** Van Dam and Zegeling, 2006 (Abstract + PDF).

**An illustrative movie (Quicktime):** mhd15ops_staircasing.mov (212KB)

**Mesh history plots:**

(Click image for larger version)

- Detail of early stage (
*t<0.12*)

(Click image for larger version)

### More examples

More examples can be found in 'Van Dam and Zegeling, 2006' (

Abstract + PDF). These include shock tubes, Shear-Alfvén waves and pseudoconvergence from critical to regular solutions.

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