PaScaL_TDMA: A library of parallel and scalable solvers for massive tridiagonal systems

The aim of this study is to devise an efficient and scalable computational procedure to solve the many tridiagonal systems in multi-dimensional partial differential equations. The modified Thomas algorithm and a newly designed communication scheme were used to reduce the communication overhead encountered while solving the many tridiagonal systems. Benchmark test results reveal an advantage of the proposed procedures compared to global all-to-all communication methods — a significantly reduced communication time that becomes more prominent for larger problem sizes and greater number of cores. The proposed computational procedures are fully implemented in an open-source library called Parallel and Scalable Library for TDMA (PaScaL_TDMA). Considering a three-dimensional heat conduction problem as a practical example, we obtain good strong and weak scalability results up to 262,144 computing cores on the KISTI Nurion cluster system, which, to the best of our knowledge, is the largest parallel simulation for solving tridiagonal systems. The potential of this library for large-scale substantive problems in physics is also demonstrated through direct numerical simulations of the Rayleigh–Bénard convection problem, which yielded excellent scalability and accurate results.