An investigation of GPU-based stiff chemical kinetics integration methods

Abstract

A fifth-order implicit Runge-Kutta method and two fourth-order exponential integration methods equipped with Krylov subspace approximations were implemented for the GPU and paired with the analytical chemical kinetic Jacobian software pyJac. The performance of each algorithm was evaluated by integrating thermochemical state data sampled from stochastic partially stirred reactor simulations and compared with the commonly used CPU-based implicit integrator CVODE. We estimated that the implicit Runge-Kutta method running on a single Tesla C2075 GPU is equivalent to CVODE running on 12-38 Intel Xeon E5-4640 v2 CPU cores for integration of a single global integration time step of 10 −6 s with hydrogen and methane kinetic models. In the stiffest case studied-the methane model with a global integration time step of 10 −4 s-thread divergence and higher memory traffic significantly decreased GPU performance to the equivalent of CVODE running on approximately three CPU cores. The exponential integration algorithms performed more slowly than the implicit integrators on both the CPU and GPU. Thread divergence and memory traffic were identified as the main limiters of GPU integrator performance, and techniques to mitigate these issues were discussed. Use of a finite-difference Jacobian on the GPU-in place of the analytical Jacobian provided by pyJac-greatly decreased integrator performance due to thread divergence, resulting in maximum slowdowns of 7.11-240.96 ×; in comparison, the corresponding slowdowns on the CPU were just 1.39-2.61 ×, underscoring the importance of use of an analytical Jacobian for efficient GPU integration. Finally, future research directions for working towards enabling realistic chemistry in reactive-flow simulations via GPU/SIMT accelerated stiff chemical kinetics integration were identified.

Using SIMD and SIMT vectorization to evaluate sparse chemical kinetic Jacobian matrices and thermochemical source terms

“…The OpenMP code-generation is currently only capable of parallel execution, but extending this platform to shallow/deep-vectorizations (via loo.py and compiler #pragmas) is a key priority going forwards since OpenMP is a standard library on most machines. In addition, CUDA [51] has been significantly more reliable in previous works [18,24], while Intel's opensource OpenCL alternative, ISPC [78], has been relatively stable and easy to work with during preliminary efforts with the unit-testing discussed in Section 2.3. The current deep-vectorization formulation would be executable for both CUDA and ISPC targets, as these languages implement doubleprecision atomic operations, further recommending their use.…”

“…In addition,~9 × and~25 × speedups were achieved for the simulation of a partially premixed methanol flame for solving partial differential equations (PDE) on 16 CPU cores and the GPU, respectively. Curtis et al [24] implemented a fifth-order implicit Runge-Kutta method [44], as well as two fourth-order exponential integration techniques [45,46] paired with an analytical Jacobian code [18] on the GPU and CPU. The GPU-based implicit Runge-Kutta method performed equivalently to a standard implicit integrator [47] running on 12-38 CPU cores for two relatively small chemical models with an integration time step of 10 −6 s.…”

“…Shallow vectorization may also experience SIMD waste or SIMT thread divergence, e.g., in chemical kinetic integration due to varying internal solver time-step sizes [24]. However, in this work shallow vectorization is largely unaffected by this concern as the only major code paths that differ between vector lanes are high/low-temperature polynomial evaluations and differing pressures for P-Log reactions, which cause far fewer issues compared with differing internal ODE integration time-steps [24]. Finally, Fig.…”

“…OpenMP Finally, Table 4 displays the chemical kinetic models used in this work, as well as number of partially stirred reaction conditions (PaSR) used in the condition database for each. Our previous works describe the creation of the PaSR databases in detail works [18,24].…”

Non-feedback technique to directly control multistability in nonlinear oscillators by dual-frequency driving

“…A thorough performance analysis between GPUs and CPUs solving large numbers of ordinary differential equations is beyond the scope of the present study. The interested reader is referred to the publications [38,[50][51][52] for more details.…”