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:

• Overview (t<=2.0)
  

(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.