An efficient preconditioning technique using Krylov subspace methods for 3D characteristics solvers

Abstract

Abstract The Generalized Minimal RESidual ( GMRES ) method, using a Krylov subspace projection, is adapted and implemented to accelerate a 3D iterative transport solver based on the characteristics method. Another acceleration technique called the self-collision rebalancing technique ( SCR ) can also be used to accelerate the solution or as a left preconditioner for GMRES . The GMRES method is usually used to solve a linear algebraic system ( A x = b ) . It uses K ( r ( o ) , A ) as projection subspace and A K ( r ( o ) , A ) for the orthogonalization of the residual. This paper compares the performance of these two combined methods on various problems. To implement the GMRES iterative method, the characteristics equations are derived in linear algebra formalism by using the equivalence between the method of characteristics and the method of collision probability to end up with a linear algebraic system involving fluxes and currents. Numerical results show good performance of the GMRES technique especially for the cases presenting large material heterogeneity with a scattering ratio close to 1. Similarly, the SCR preconditioning slightly increases the GMRES efficiency.