Solution of single linear tridiagonal systems and vectorization of the ICCG algorithm on the Cray 1

Abstract

The numerical algorithms used to solve the physics equation in codes which model laser fusion are examined, it is found that a large number of subroutines require the solution of tridiagonal linear systems of equations. One dimensional radiation transport, thermal and suprathermal electron transport, ion thermal conduction, charged particle and neutron transport, all require the solution of tridiagonal systems of equations. The standard algorithm that has been used in the past on CDC 7600's will not vectorize and so cannot take advantage of the large speed increases possible on the Cray-1 through vectorization. There is however, an alternate algorithm for solving tridiagonal systems, called cyclic reduction, which allows for vectorization, and which is optimal for the Cray-1. Software based on this algorithm is now being used in LASNEX to solve tridiagonal linear systems in the subroutines mentioned above. The new algorithm runs as much as five times faster than the standard algorithm on the Cray-1. The ICCG method is being used to solve the diffusion equation with a nine-point coupling scheme on the CDC 7600. In going from the CDC 7600 to the Cray-1, a large part of the algorithm consists of solving tridiagonal linear systems on each L linemore » of the Lagrangian mesh in a manner which is not vectorizable. An alternate ICCG algorithm for the Cray-1 was developed which utilizes a block form of the cyclic reduction algorithm. This new algorithm allows full vectorization and runs as much as five times faster than the old algorithm on the Cray-1. It is now being used in Cray LASNEX to solve the two-dimensional diffusion equation in all the physics subroutines mentioned above.« less