Abstract

Large-scale sparse linear equation system need be solved repeatedly in many scientific simulations. The iterative solvers are more preferable than direct solvers due to their lower computation cost and memory requirement. The Krylov iteration methods are typically more effective than stationary iteration method.

The Krylov solvers need a good preconditioner to improve their efficiency.  A preconditioner is a matrix close to the original matrix which can be easily constructed and the linear system with this matrix should be easily solved. For serial computing, ILU, SOR is widely used as efficient preconditioners. For parallel computing, the connections between process are generally dropped in preconditioners, Block Jacobian with ILU within block are used as the best preconditioner in linear algebraic software packages, such as PETSc.  SOR can be used as a preconditioner in parallel with multi-color ordering, but it is tedious to implement.

By analysis the SOR with red-black coloring preconditioner, we propose a reduced SOR preconditioner RSOR which do not need multi-color ordering and can be easily implemented in parallel computing. We further improved the preconditioner by replacing SOR with ILU within each process, so we obtained a hybrid preconditioner, RSOR for the connections between processes and ILU within process, called RSILU.

Both preconditioners, RSOR and RSILU, have been implemented with parallel Krylov solver, GMRES, and applied to solve large sparse linear systems of CMFD for 3D whole core neutronics simulation. The numerical experiments show that RSOR is more efficient than BJILU for a large number of processes, and RSILU is better than RSOR in all cases.

Bio

Dr. Yunlin Xu had received his bachelor’s degree in 1990 and a master’s degree in 1991, both from Tsinghua University and received his Ph. D from Purdue in 2004, all majored in Nuclear Engineering. He had also received a master’s degree in computer science from Purdue in 2006.

He was an associate professor at Tsinghua University from 1991 to 1999. his research was in the area of nuclear reactor physics design and safety analysis. He was responsible for the reactor physics design of two advanced nuclear reactors: a 200 MW nuclear heating reactor (NHR) and a 10 MW high-temperature gas-cooled reactor. After he came to Purdue University at end of 1999, his research area has switched to methods and code development for Neutron kinetics and thermal-fluid coupling simulation, multi-physics coupling.  At 2008, Dr. Xu left Purdue to join Argonne National Laboratory but worked off campus at University of Michigan. 10 years later, Dr. Xu came back to Purdue as a faculty member of the School of Nuclear Engineering. His research interest also includes large linear and non-linear system solution methods. He was primary developer of neutronics codes such as PARCS, DeCart, MPACT. He also contributes to nuclear reactor system codes such as TRACE, RELAP. His researches are shown in more than 80 published papers.