Title of program: ILUCG2 (Incomplete LU factorized Con-being stiff and requiring implicit solution techniques. Generjugate Gradient algorithm for 2D problems) ally, the resulting matrix equations are asymmetric; we solve them here with the ILUCG2 program. In a subsequent article Catalogue number: ACEU we describe a simpler and faster algorithm, ICCG2, which should be used when the matrix is symmetric [4). Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland (see application form in this issue) Method of soltion A generalization of the incomplete Cholesky conjugate gradient Computer: Cray-I; Installation: NMFECC, Livermore (ICCG) algorithm is used to solve the linear asymmetric matrix equation [5,6). Programming language used: FORTRAN Restrictions on the complexity of the problem Operating system: CTSS The discretization ofthe two-dimensionalPDE and its boundary conditions must result in a spatial 9-point operator stencil High speed store required: at least 22 * mn where mn is the which can be represented by a block tridiagonal matrix cornnumber of linear equations posed of tridiagonal blocks. Number of bits in a word: 64 Typical running times These are problem dependent because ill-conditioned matrices Peripheral used: printer will require more iterations than well conditioned ones. However, the running times per iteration are known. For enough Number of card images in combined program and test file: 1193 equations (1189 here) the time for the incomplete LU factoriza-(599 and 594 in the general source and standard FORTRAN tion is 5.9 ss per unknown. Each conjugate gradient iteration source, respectively) requires 4.2 ~s per unknown. In the three test problems the number of iterations required to reach a relative residual error Card punching code: 8 bit ASCII of 10 1~(in the L2 norm) ranged from 9 to 208. Unusual features of the program Keywords: partial differential equations, elliptic, parabolic,
The loops are arranged to vectorize on the Cray-I (with the two-dimensional, plasma physics, implicit, preconditioned, con-CFT compiler) wherever possible. Recursive loops (which canjugate gradient algorithm, ICCG, ILUCG, asymmetric matrix, not be vectorized) are written for optimum scalar speed. Some non-symmetnc matrix of these loops were made more efficient by increasing the arithmetic per pass and at the same time reducing the number Nature of the physical problem of passes. This trick significantly reduced loop overhead and Elliptic and parabolic partial differential equations which arise made these scalar loops 30% faster. in plasma physics applications (as well as in others) are solved in two dimensions. Plasma diffusion, equilibria and phase space References transport (Fokker-Planck equation) have been treated by these
[I] R.W. Harvey, M.G. McCoy, J.Y. Hsu and A.A. Mirin,.These problems share the common feature of Phys. Rev. Lett. 47 (1981) 102.