The use of a preconditioned bi-conjugate gradient method for hybrid plasma stability analysis

Abstract The numerical stability analysis of compact toroidal plasmas using implicit time differencing requires the solution of a set of coupled, 2-dimensional, elliptic partial differential equations for the field quantities at every timestep. When the equations are spatially finite-differenced and written in matrix form, the resulting matrix is large, sparse, complex, non-Hermitian, and indefinite. The use of the preconditioned bi-conjugate gradient method for solving these equations is discussed. The effect of block-diagonal preconditioning and of incomplete block-LU preconditioning on the convergence of the method is investigated. For typical matrices arising in our studies, the eigenvalue spectra of the original and preconditioned matrices are calculated as an illustration of the effectiveness of the preconditioning. We show that the preconditioned bi-conjugate gradient method converges more rapidly than the conjugate gradient method applied to the normal equations, and that it is an effective iterative method for the class of non-Hermitian, indefinite problems of interest.

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