An adaptive implicit time-integration scheme for stiff chemistry based on Jacobian tabulation method

The use of detailed chemical mechanisms in modeling chemically reacting flows often results in large computational costs. In this study, a stiff ODE solver, combining the backward differentiation formula (BDF) with a novel Jacobian tabulation (JT) method, was developed to significantly reduce the computational cost of solving chemical ordinary differential equations (ODEs) with detailed chemistry. The resulting BDF-JT method offers several key advantages: (a) it accelerates the evaluation and factorization of the Jacobian matrix during the BDF solution process through a two-level dynamic tabulation; (b) a properly selected set of state parameters governs the identification of similar Jacobians and their factorizations within the tabulation system; and (c) efficient look-up and dynamic updates of the tabulation are enabled using hash tables. The accuracy and efficiency of the BDF-JT method were systematically compared against the original BDF and CVODE solvers across a range of scenarios, including homogeneous auto-ignition, laminar flame propagation, and oblique detonation waves (ODW). The results showed that the proposed method had negligible effects on accuracy under the studied tolerances. In terms of computational efficiency, the BDF-JT method achieved significant speedup, reducing the computational cost of solving chemical ODEs by a factor of 173.7 in the auto-ignition case with a 2885-species mechanism and by a factor of 41.3 for ODW combustion with a 1384-species mechanism, with higher efficiency observed for more complex reaction systems. Overall, the BDF-JT method demonstrates strong potential for drastically reducing computational costs while maintaining high accuracy in large-scale combustion simulations, making it a valuable tool for high-fidelity combustion simulation.

Novelty and Significance Statement

There is an urgent need to incorporate more realistic chemical mechanisms in modeling chemical reacting flows. Due to the large computational cost of using detailed chemical kinetics, operator-splitting schemes along with several stiff ODE solvers have been studied and demonstrated performance. Unfortunately, most stiff ODE solvers are likely developed trading off accuracy for performance. The present study provides a dynamic tabulation method for chemical Jacobian in the implicit ODE solvers. As such, it helps to accelerate the computations in combustion simulations without sacrificing accuracy.