Parallelizing multiple precision Taylor series method for integrating the Lorenz system

TL;DR

This paper presents a hybrid MPI+OpenMP parallelization of the high-precision Taylor series method for solving the Lorenz system, achieving accurate long-term solutions using extensive computational resources.

Contribution

It introduces a novel hybrid parallelization approach combining MPI and OpenMP with GMP libraries for high-precision Taylor series integration.

Findings

Successfully computed long-term solutions of the Lorenz system with 3374-3510 decimal digits precision.

Achieved a speedup of approximately 105 using 192 CPU cores.

Demonstrated the method's effectiveness for high-precision, long-time integration.

Abstract

A hybrid MPI+OpenMP strategy for parallelizing multiple precision Taylor series method is proposed, realized and tested. To parallelize the algorithm we combine MPI and OpenMP parallel technologies together with GMP library (GNU miltiple precision libary) and the tiny MPIGMP library. The details of the parallelization are explained on the paradigmatic model of the Lorenz system. We succeed to obtain a correct reference solution in the rather long time interval - [0,7000]. The solution is verified by comparing the results for 2700-th order Taylor series method and precision of ~ 3374 decimal digits, and those with 2800-th order and precision of ~ 3510 decimal digits. With 192 CPU cores in Nestum cluster, Sofia, Bulgaria, the 2800-th order computation was ~ 145 hours with speedup ~ 105.