Chaotic linear equation-system solvers for unsteady CFD

Abstract

A Chaotic Iterative Method, which is a form of totally asynchronous linear

equation-system solver, is implemented within an open-source framework. The solver is similar to simple Jacobi or Gauss-Seidel methods, but is highly optimized for massively- parallel computations. Processes or threads are free to run computations regardless of the current state of other processes, iterating individual equations with no limitations on the state of the variables which they use. Each individual iteration may pull variables from the same iteration, the previous iteration, or indeed any iteration. This effectively removes all synchronization from the Jacobi or Gauss-Seidel algorithm, allowing computations to run efficiently with high concurrency.

The trade-off is that the numerical convergence rate of these simple algorithms is slower compared to the classical Krylov Subspace methods, which are popular today. However, unique features of the computational fluid dynamics algorithm work in favour of Chaotic methods, allowing the fluid dynamics field to exploit these algorithms when other’s cannot. The results of the Chaotic solver are presented, verifying the numerical results and benchmarking performance against the Generalized Minimal Residual (GMRES) solver and a Pipelined GMRES solver. The results show that, under certain circumstances, Chaotic methods could be used as a standalone solver due to their superior scalability. The potential to use Chaotic methods as a pre-conditioner or hybrid solver is also revealed.