Nonlinear Magnetohydrodynamics with Finite Elements
Carl Sovinec
Plasma Theory Group (T-15)
Los Alamos National Laboratory
and the
NIMROD Team
presented to the
Numerical Analysis Seminar
February 9, 2000
OUTLINE
The NIMROD Code:
The NIMROD Team:
NIMROD FEATURES
One reason for avoiding a potential representation is to maintain high convergence order on the physical quantities. Since derivatives of the Galerkin quantities lose one order of accuracy for each derivative, convergence of B may be slow if a code solves for A, where .
The discrete divergence integral (the numerator of the div-stab ratio) will be set to 0 in the finite element solution. The question this condition answers is, "Does that give you a divergence free vector field in the limit of h->0?"
The advance of time represents a mapping of the magnetic field. In the numerical case with elements not satisfying divergence-stability, part of the mapping from generates non-solenoidal field. Similarly, part of the mapping from error diffusion generates unphysical contributions to the solenoidal field.
If the generation of non-solenoidal field is not too strong, setting k_{b}~h is sufficient. Then, the physical resistive diffusion is sufficient to dissipate unphysical solenoidal field generated by the error diffusion.
Having k_{b}~h seems to be insufficient for cases with strong (ion or electron) flow.
A linear tearing-mode (exponential growth with time) serves as a test for divergence error and convergence.
The geometry is a circular cross section, periodic cylinder. The polar grid is composed of quadrilaterals.
We have varied k_{b} over orders of magnitude to compare bilinear (32x32 mesh) and biquadratic solutions (16x16 mesh). (h=1000)
Linear Growth Rate vs. log_{10}(k_{b})
Divergence Error, , vs. log_{10}(k_{b})
In a similar, but doubly-periodic slab, using k_{b}=50h was catastrophic for bilinear elements:
Repeating the computation with k_{b}=h avoided the vertex-to-vertex oscillations.
Convergence of the solution and error are improved using biquadratic elements in the cylindrical problem.
Radial and azimuthal resolution vary simultaneously. [k_{b}=2h]
Convergence of the solution and error are compared for the cylindrical problem, varying only the number of cells in the azimuthal direction. [k_{b}=2h]
SUMMARY