Nonlinear Magnetohydrodynamics with Finite Elements

Carl Sovinec

Plasma Theory Group (T-15)

Los Alamos National Laboratory

and the



presented to the

Numerical Analysis Seminar

February 9, 2000









The NIMROD Code:

The NIMROD Team:









The NIMROD spatial representation is finite element for one plane and Fourier series for the third (periodic) direction.







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 kb~h is sufficient. Then, the physical resistive diffusion is sufficient to dissipate unphysical solenoidal field generated by the error diffusion.

Having kb~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.


Equilibrium B


Linear Eigenfunction ~




We have varied kb over orders of magnitude to compare bilinear (32x32 mesh) and biquadratic solutions (16x16 mesh). (h=1000)

Linear Growth Rate vs. log10(kb)

Divergence Error, , vs. log10(kb)




In a similar, but doubly-periodic slab, using kb=50h was catastrophic for bilinear elements:



Repeating the computation with kb=h avoided the vertex-to-vertex oscillations.




Convergence of the solution and error are improved using biquadratic elements in the cylindrical problem.

Linear Growth Rate vs. Mesh Spacing

ln(Divergence Error) vs. ln(Mesh Spacing)

Radial and azimuthal resolution vary simultaneously. [kb=2h]




Convergence of the solution and error are compared for the cylindrical problem, varying only the number of cells in the azimuthal direction. [kb=2h]

Linear Growth Rate vs. Mesh Spacing

ln(Divergence Error) vs. ln(Mesh Spacing)