Application of preconditioned GMRES to the numerical solution of the neutron transport equation

The generalized minimal residual (GMRES) method with right preconditioning is examined as an alternative to both standard and accelerated transport sw eeps for the iterative solution of the diamond differenced discrete ordinat es neutron transport equation. Incomplete factorization (ILU) type precondi tioners are used to determine their effectiveness in accelerating GMRES for this application. ILU(tau), which requires the specification of a dropping criteria tau, proves to be a good choice for the types of problems examine d in this paper. The combination of ILU(tau) and GMRES is compared with bot h DSA and unaccelerated transport sweeps for several model problems. It is found that the computational workload of the ILU(tau)-GMRES combination sca les nonlinearly with the number of energy groups and quadrature order, maki ng this technique most effective for problems with a small number of groups and discrete ordinates. However, the cost of preconditioner construction c an be amortized over several calculations with different source and/or boun dary values. Preconditioners built upon standard transport sweep algorithms are also evaluated as to their effectiveness in accelerating the convergen ce of GMRES. These preconditioners show better scaling with such problem pa rameters as the scattering ratio, the number of discrete ordinates, and the number of spatial meshes. These sweeps based preconditioners can also be c ast in a matrix free form that greatly reduces storage requirements. (C) 20 01 Elsevier Science Ltd. All rights reserved.