A box-shaped cyclically reduced operator

Abstract

We propose a new procedure of partial cyclic reduction, where we apply a 2 d -color ordering (with d = 2, 3 the dimension of the problem), and use different operators for different gridpoints according to their color. These operators are chosen so that the gridpoints can be readily decoupled, and we then eliminate all colors but one. This yields a smaller cartesian mesh and box-shaped 9-point (in 2D) or 27-point (in 3D) operators that are easy to analyze and implement. Multi-line and multi-plane orderings are considered, and we perform convergence analysis and numerical experiments that demonstrate the merits of our approach.