A Cylindrical Arbitrary Lagrangian Eulerian MHD Code
T.Goffrey
Centre for Fusion, Space and Astrophysics, Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
t.goffrey@warwick.ac.uk
A two dimensional r-z Arbitrary Lagrangian
Eulerian MHD code is under development
in order to study problems with such a
symmetry. A number of one dimensional
tests to illustrate some of the problems
and possible solutions are presented.
•Lagrangian schemes use a grid that
moves with the fluid, Eulerian grids are
stationary
•An ALE code attempts to combine the
advantages of both, whilst avoiding the
weaknesses
•Capture of shocks is of great interest, one
such method involves the addition of
shock viscosity to the normal pressure in
the momentum and energy update.
•The Eulerian scheme presented uses a
Richtmeyer scheme, a variant of the Lax
Wendroff family of schemes, the
Lagrangian is a staggered predictor
corrector scheme.
Solution to Sod's shock tube problem at t= 15
carried out with an Eulerian solver and bulk
shock viscosity. Under and overshoot are
clearly noticeable at multiple points, and the
shock is spread over several points.
•Both conserve energy exactly, in two
dimensions the ALE code uses a
compatible energy update to exactly
preserve energy.
•The final equations for MHD in
Lagrangian terms, including the shock
viscosity, q are:
Solution to Sod's shock tube problem using a
Lagrangian scheme with monotonic shock
viscosity. Oscillations behind the shock are
now completely removed. Resolution at the
shock front is improved.
Solution to Sod's shock tube problem at t = 15,
using a Lagrangian solver with bulk shock
viscosity. Oscillations are reduced and
resolution improved compared to that of the
Eulerian scheme.
Presented are five different
solutions to Sod's shock tube[2].
This one dimensional problem
consists initially of two ideal
gases initially at rest separated
by a diaphragm. The gas to the
left of the diaphragm, the driver
initially has uniform density and
pressure. The test section has
density of 0.125, pressure of 0.1.
After t=0 the diaphragm is
assumed to have no further
influence. The analytic solution is
plotted in a solid line whilst the
numerical solution is dashed.
Clockwise from top left, density,
velocity, energy and pressure.
Units are arbitrary, but for a driver
pressure of 1Mp and density of
1.0g/cc, the final time
corresponds to 15 microseconds
Solution to Sod's shock tube at t = 15, using a
Lagrangian remap scheme, remapping to the
original time step, without shock viscosity.
Some oscillations are present, but resolution is
in parts improved using this scheme.
Solution to Sod's shock tube problem at t = 15,
using a Lagrangian remap scheme remapping
to the original grid, using monotonic shock
viscosity. The overshoot at rarefaction is
reduced, and resolution improved compared to
the purely Lagrangian scheme.
The main choice to be made in terms of remapping is where to.
●
Presented here are two options, firstly back to the original grid “Eulerian grid motion”, or
back to a newly created equipotential grid.
●
The equipotential grid can be calculated when needed by inverting Laplace's equation in
terms of position and grid coordinate potential.
●
The new grid can be calculated over multiple iterations. Here only one per remap is used.
This reduces departure from Lagrangian grid motion
●
Should the Lagrangian grid be orthogonal, the equipotential grid will be to. If the Lagrangian
grid is close to orthogonal, the equipotential grid will move closer.
●
It is also possible to attach weighting functions, to each node, to increase resolution in areas
of interest. These can be user, or automatically defined.
●
To illustrate the difference between the two grids, two results from the interacting blast waves
of Colella and Woodward are presented [4].]Uniform weighting was given to all nodes, and a
remap in each case was performed at each time step.
●
Density profile from the interacting blast wave
problem, using bulk shock viscosity and
remapping back to the original grid. This was
found to be the same as the solution obtained
with an established code, LARE.
Density profile from the interacting blast wave problem,
using bulk shock viscosity and remapping back to an
equipotential grid. Comparisons to an established code,
LARE[1], show that for Eulerian grid motion, approximately
6 times the resolution was needed to match this solution.
Results from the Brio and Wu test problem
carried out using a Lagrangian remap scheme
remapping back to the original grid, and using
a bulk shock viscosity.
Introducing a magnetic field into the
problem further complicates matters. As
with kinetic energy, there is a
discrepancy in magnetic energy
conservation It is also imperative to
maintain the divergence free nature of
the magnetic field during remap, this is
trivial in one dimension, but not so in
higher dimensions. It is also likely for
optimal results a tensor shock viscosity
will be needed. Presented are results
from the problem first proposed by Brio
and Wu [3]. In a magnetised version of
the shock tube, the hydrodynamic initial
conditions are unchanged, but a
uniform magnetic field in the xdirection, and a discontinuous ycomponent are introduced. Clockwise
from top left are density, velocity in the
x-direction, the y component of the
magnetic field and the velocity in the ydirection
Results from the Brio and Wu test case
using a Lagrangian remap scheme
remapping back the the original grid and
using monotonic shock viscosity.
Results for the Brio and Wu test case
using a Lagrangian remap scheme
remapping to an equipotential grid and
using a monotonic shock viscosity.
[1] T. D. Arber et al. J. Comp. Phys. 171 (2001) 151
[2] G. A. Sod. J. Comp. Phys. 27 (1978,1
[3] M. Brio and C.C.Wu. J. Comp. Phys. 75(1988) 400
[4] P. Woodward and P. Colella. J. Comp. Phys. 54(1984) 115
This work acknowledges the financial support of AWE