SIMD vectorization for simultaneous solution of locally varying linear systems with multiple right-hand sides

Abstract

Developments in numerical simulation of flows and high-performance computing influence one another. More detailed simulation methods create a permanent need for more computational power, while new hardware developments often require changes to the software to exploit new hardware features. This dependency is very pronounced in the case of vector-units which are featured by all modern processors to increase their numerical throughput but require vectorization of the software to be used efficiently. We study the vectorization of a simulation method that exhibits an inherent level of vector-parallelism. This is of particular interest as SIMD operations will hopefully be available with std::simd in a future C++ standard. The simulation method considered here results in the simultaneous solution of multiple sparse linear systems of equations which only differ by their main diagonal and right-hand sides. Such structure arises in the simulation of unsteady flow in turbomachinery by means of a frequency domain approach called harmonic balance.

Vectorized Numerical Algorithms to Solve Internal Problems of Computational Fluid Dynamics

“…Internal cells are highlighted in bold in Figure 2, and addressing them is provided using the index vector pC. pC={7, 8,9,12,13,14,17,18,19,22,23,24,27,28 The method of addressing ghost cells is shown in Figure 3. For this purpose, index vectors pBL, pBU, pBR, pBD are designed, corresponding to left, upper, right, and lower boundaries of computational domain.…”

“…There are many examples of vectorized numerical algorithms that enable improving performance of code running on modern CPUs. These examples include adaptive quadrature codes process [15], solution of linear algebraic equations [16,17], integration of ordinary differential equations [18], and convergence acceleration techniques [19]. Applications of various vectorized algorithms for solving CFD problems are reported in [20][21][22][23][24].…”