Adapting Chebyshev Iteration for Solving the AORSA Matrix*

The “All Orders” Larmor Radius solve for ICRF in hot plasmas, e.g., the AORSA code[1] and its predecessor, the METS code[2], produce a linear system of equations which is complex-valued, non-symmetric, non-definite, with multiple physical modes of different wavelengths, that makes effective pre-conditioning difficult-to-impossible.  Thus, this challenging system is presently solved by direct matrix inversion.  However, the N-cubed operation-count of direct matrix inversion restricts the problem size, to iterated 2D, e.g., the axi-symmetric tokamak, but not for the stellarator, or for a non-axi-symmetric perturbation within a tokamak.  In order to solve a true non-axi-symmetric 3D problem, an alternate solve technique is required, which we demonstrate.  We have adapted Chebyshev accelerated Richardson iteration for use with linear systems having spectra with eigenvalues of both positive and negative real part, and zero or negative imaginary part.  We use traditional Richardson iteration, with each iterate providing a different polynomial root, instead of using the Chebyshev recursion formulae. Roundoff is very well controlled with a fractal-based resequencing of the root order.[3]  This approach allows us to modify the Chebyshev polynomial by making small changes to the root locations, and also allows us to use a “semi-ellipse” polynomial, where the roots lie on a half ellipse, instead of a line.  The combination allows for a half-ellipse spectral hull, which can touch and extend to either side of the origin, thus significantly generalizing the class of non-symmetric problems that can be treated with Chebyshev Iteration, including the AORSA matrix.

[1] E.F. Jaeger, L.A. Berry, E. D'Azevedo, et al., Phys. Plasmas. 8, 1573 (2001).

[2] D.N. Smithe, Plasma Phys. Control. Fusion, 31 (1989) 1105.

[3] David Smithe, Larry Ludeking, Comp. Phys. Comm. 106, 95 (1997).