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This study presents an analytical Jacobian formulation for detailed gas-phase reaction kinetics, suitable for accurate and computationally efficient combustion simulations using either skeletal or detailed reaction mechanisms. A general chemical kinetics initial value problem in constant volume environments is considered, where the gas-phase mixture thermodynamic properties are polynomial functions of temperature according to the JANAF standard. Three different reaction behaviors are accounted for, including modified Arrhenius kinetic law reactions, third-body collisions, and pressure-dependent reactions in Lindemann's or Troe's kinetic law forms. The integration of the chemistry ordinary differential equations (ODE) system is carried out using a software package specifically developed in Fortran language, and the solution is compared to a reference chemical kinetics library. Two analytical Jacobian formulations, an exact one and a sparser approximate one, are proposed and compared to numerical Jacobians computed by finite differences internally generated by a variety of commonly used stiff ODE solvers. The results show significant reductions in total computational times for the chemistry ranging from factors of 2 to more than 2 orders of magnitude for 29 species, 56 reactions to 2878 species, 8555 reactions, respectively. Finally, the code has been coupled to an engine combustion simulation software, where at each time step the chemistry ODE system is integrated in each cell of the computational grid, allowing 77% faster computations with a 160 species combustion mechanism.
Perini et al. (Wed,) studied this question.