An analytical Jacobian approach to sparse reaction kinetics for computationally efficient combustion modelling with large reaction mechanisms

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 problemin 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 behaviours 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 ODE system is carried out using a software package specifically developed in Fortran language, and the solution 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 ordinary differential equations (ODE) solvers. The results show significant reductions in total computational times for the chemistry ranging from factors of 2 to more than two 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 timestep the chemistry ODE system is integrated in each cell of the computational grid, allowing 77% faster computations with a 160 species combustion mechanism.