A parallel Jacobi-Davidson method for solving generalized eigenvalue problems in linear 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. The computation of an approximate solution of a large system of linear equations is usually the most expensive step in the algorithm. By using a suitable preconditioner, only a moderate number of steps of an inner iteration is required in order to retain fast convergence for the JD process. Several preconditioning techniques are discussed. It is shown, that for our application, a proper preconditioner is a complete block LU decomposition, which can be used for the computation of several eigenpairs. Reordering strategies based on a combination of block cyclic reduction and domain decomposition result in a well-parallelizable preconditioning technique. Results obtained on 64 processing elements of both a Cray T3D and a T3E will be shown.