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

• Introduction
• • Physics problems
  • Time-dependent equations
• Established analysis for incompressible, stationary Navier-Stokes
• • 3 approaches
  • Divergence-stability
  • Spatial convergence
• Time-dependent problems
• • Error diffusion
  • Test results
• Summary

The NIMROD Code:

•  

• Developed to simulate nonlinear electromagnetic phenomena that fundamentally alter the magnetic confinement of fusion-grade plasmas.

•  

• Designed to handle the stiff mathematical equations that describe hot magnetized plasmas, where time and spatial scales vary by many orders of magnitude.

•  

• Implemented in a parallel algorithm to run on the fastest computers available today and in the future.

•  

• NIMROD is publicly available from http://nimrodteam.org

The NIMROD Team:

•  

• NIMROD is a multi-institutional project commissioned by the Department of Energy's Office of Fusion Energy Science. The product is publicly available to provide the greatest possible benefit to the fusion community.

•  

• The NIMROD Team presently includes members from SAIC-San Diego, SNL, UT-Austin, General Atomics, UCLA, CU-Boulder, MSU, and LANL.

NIMROD FEATURES

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

•  

• Permits simulating detailed configurations for comparison with experiment and for simulating non-fusion applications.

•  

• Allows refinement in regions of interest.

•  

• Present and future developments will take advantage of the flexibility of finite elements to change basis functions and to dynamically adapt the mesh to solutions.

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.

Equilibrium B

Linear Eigenfunction ~

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. log₁₀(k_b)

Divergence Error, , vs. log₁₀(k_b)

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

•  

• A slow numerical mode arose after 0.15 diffusion times.

•  

• There is a mode in the envelope of the eigenfunction.

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.

Linear Growth Rate vs. Mesh Spacing

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

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]

Linear Growth Rate vs. Mesh Spacing

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

SUMMARY

•  

• Published numerical analysis of finite elements applied to the incompressible Navier-Stokes equations has proven extremely valuable for understanding and improving the advance of magnetic field in our nonlinear plasma simulation code.

•  

• Our original formulation with linear and bilinear elements did not satisfy the divergence-stability condition.

•  

• The error diffusion technique with k_b~h works with low-order elements that do not satisfy divergence-stability in cases where the generation of is easily controllable.

•  

• The divergence-stability condition is satisfied with quadratic elements and the error diffusion technique. The rate of converge is increased, and results are much less sensitive to the value of k_b.