A Parallel Jacobi--Davidson-type Method for Solving Large Generalized Eigenvalue Problems in Magnetohydrodynamics

We study the solution of generalized eigenproblems generated by a model which is used for stability investigation of tokamak plasmas. The eigenvalue problems are of the form $A x = \lambda B x$, in which the complex matrices A and B are block-tridiagonal, and B is Hermitian positive definite. The Jacobi--Davidson method appears to be an excellent method for parallel computation of a few selected eigenvalues because the basic ingredients are matrix vector products, vector updates, and inner products. The method is based on solving projected eigenproblems of order typically less than 30. We apply a complete block LU decomposition in which reordering strategies based on a combination of block cyclic reduction and domain decomposition result in a well-parallelizable algorithm. One decomposition can be used for the calculation of several eigenvalues. Spectral transformations are presented to compute certain interior eigenvalues and their associated eigenvectors. The convergence behavior of several variants of the Jacobi--Davidson algorithm is examined. Special attention is paid to the parallel performance, memory requirements, and prediction of the speed-up. Numerical results obtained on a distributed memory Cray T3E are shown.