The Chebychev method for solving nonself-adjoint elliptic equations on a vector computer

Abstract

The Chebychev explicit method can be extended to nonsymmetric operators L whose complex eigenvalues lie within an ellipse in the complex plane. The vectorizability of the method results in high execution efficiency on a “pipeline” computer. We derive the method and its convergence rate, and give a comparison with two other methods. The comparison is taken from a 2D plasma turbulence code, in which L = ▽ 2 + A( x, y) · ▽. The explicit method is approximately three times more efficient than ADI for the model problem solved on a two-pipe Texas Instruments ASC. In some cases, a staggered mesh can be used to gain another factor of 2 in the efficiency of the explicit method. The method has been used successfully on meshes of 34 × 34, 50 × 50, and 130 × 130 points. For grids of 50 or more points on a side, we show in the Appendix that convergence can be speeded considerably by the use of a suitably chosen auxiliary coarse grid, on which long-wavelength components of the error are corrected.