Accurate simulations of combustion phenomena require the use of detailed chemical kinetics in order to capture limit phenomena such as ignition and extinction as well as predict pollutant formation. However, the chemical kinetic models for hydrocarbon fuels of practical interest typically have large numbers of species and reactions and exhibit high levels of mathematical stiffness in the governing differential equations, particularly for larger fuel molecules. In order to integrate the stiff equations governing chemical kinetics, generally reactive-flow simulations rely on implicit algorithms that require frequent Jacobian matrix evaluations. Some in situ and a posteriori computational diagnostics methods also require accurate Jacobian matrices, including computational singular perturbation and chemical explosive mode analysis. Typically, finite differences numerically approximate these, but for larger chemical kinetic models this poses significant computational demands since the number of chemical source term evaluations scales with the square of species count. Furthermore, existing analytical Jacobian tools do not optimize evaluations or support emerging SIMD processors such as GPUs. Here we introduce pyJac, a Python-based open-source program that generates analytical Jacobian matrices for use in chemical kinetics modeling and analysis. In addition to producing the necessary customized source code for evaluating reaction rates (including all modern reaction rate formulations), the chemical source terms, and the Jacobian matrix, pyJac uses an optimized evaluation order to minimize computational and memory operations. As a demonstration, we first establish the correctness of the Jacobian matrices for kinetic models of hydrogen, methane, ethylene, and isopentanol oxidation (number of species ranging 13-360) by showing agreement within 0.001 % of matrices obtained via automatic differentiation. We then demonstrate the performance achievable on CPUs and GPUs using pyJac via matrix evaluation timing comparisons; the routines produced by pyJac outperformed first-order finite differences by 3-7.5 times and the existing analytical Jacobian software TChem by 1.1-2.2 times on a single-threaded basis. It is noted that TChem is not thread-safe, while pyJac is easily parallelized, and hence can greatly outperform TChem on multicore CPUs. The Jacobian matrix generator we describe here will be useful for reducing the cost of integrating chemical source terms with implicit algorithms in particular and algorithms that require an accurate Jacobian matrix in general. Furthermore, the open-source release of the program and Python-based implementation will enable wide * Corresponding author

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.

The relative importance index (RII) method for determining appropriate target species for dynamic adaptive chemistry (DAC) simulations using the directed relation graph with error propagation (DRGEP) method is developed. The adequacy and effectiveness of this RII method is validated for two fuels: n-heptane and isopentanol, representatives of a ground transportation fuel component and bio-alcohol, respectively.The conventional method of DRGEP target species selection involves picking an unchanging (static) set of target species based on the combustion processes of interest; however, these static target species may not remain important throughout the entire combustion simulation, adversely affecting the accuracy of the method. In particular, this behavior may significantly reduce the accuracy of the DRGEP-based DAC approach in complex multidimensional simulations where the encountered combustion conditions cannot be known a priori with high certainty. Moreover, testing multiple sets of static target species to ensure the accuracy of the method is generally computationally prohibitive. Instead, the RII method determines appropriate DRGEP target species solely from the local thermo-chemical state of the simulation, ensuring that accuracy will be maintained. Further, the RII method reduces the expertise required of users to select DRGEP target species sets appropriate to the combustion phenomena under consideration.Constant volume autoignition simulations run over a wide range of initial conditions using detailed reaction mechanisms for n-heptane and isopentanol show that the RII method is able to maintain accuracy even when traditional static target species sets fail, and are even more accurate than expert-selected target species sets. Additionally, the accuracy and efficiency of the RII method are compared to those of static target species sets in single-cell engine simulations under homogeneous charge compression ignition conditions. For simulations using more stringent DRGEP thresholds, the RII method performs similarly to that of the static target species sets. With a larger DRGEP threshold, the RII method is significantly more accurate than the static target species sets without imposing significant computational overhead.Furthermore, the applicability of the RII method to a DRG-based DAC scheme is discussed.

Speedups reached up to 17.60 × and 45.13 × for dense and sparse evaluation on the GPU, and up to 55.11 × and 245.63 × on the CPU over a first-order finite-difference Jacobian approach. Further, dense Jacobian evaluation was up to 19.56 × and 2.84 × times faster than a previous version of pyJac on a CPU and GPU, respectively. Finally, future directions for vectorized chemical kinetic evaluation and sparse linear-algebra techniques were discussed.

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