A study of direct and Krylov iterative sparse solver techniques to approach linear scaling of the integration of chemical kinetics with detailed combustion mechanisms

The integration of the stiff ODE systems associated with chemical kinetics is the most computationally demanding task in most practical combustion simulations. The introduction of detailed reaction mechanisms in multi-dimensional simulations is limited by unfavorable scaling of the stiff ODE solution methods with the mechanism’s size. In this paper, we compare the efficiency and the appropriateness of direct and Krylov subspace sparse iterative solvers to speed-up the integration of combustion chemistry ODEs, with focus on their incorporation into multi-dimensional CFD codes through operator splitting. A suitable preconditioner formulation was addressed by using a general-purpose incomplete LU factorization method for the chemistry Jacobians, and optimizing its parameters using ignition delay simulations for practical fuels. All the calculations were run using a same efficient framework: SpeedCHEM, a recently developed library for gas-mixture kinetics that incorporates a sparse analytical approach for the ODE system functions. The solution was integrated through direct and Krylov subspace iteration implementations with different backward differentiation formula integrators for stiff ODE systems: LSODE, VODE, DASSL. Both ignition delay calculations, involving reaction mechanisms that ranged from 29 to 7171 species, and multi-dimensional internal combustion engine simulations with the KIVA code were used as test cases. All solvers showed similar robustness, and no integration failures were observed when using ILUT-preconditioned Krylov enabled integrators. We found that both solver approaches, coupled with efficient function evaluation numerics, were capable of scaling computational time requirements approximately linearly with the number of species. This allows up to three orders of magnitude speed-ups in comparison with the traditional dense solution approach. The direct solvers outperformed Krylov subspace solvers at mechanism sizes smaller than about 1000 species, while the Krylov approach allowed more than 40% speed-up over the direct solver when using the largest reaction mechanism with 7171 species.