A second-order dynamic adaptive hybrid scheme for time-integration of stiff chemistry

Abstract

A dynamic adaptive hybrid integration (AHI) scheme of second-order accuracy (AHI2) is proposed for time-integration of chemically reacting flows involving stiff chemistry. AHI2 is extended from a first-order AHI method (AHI1) developed in a previous study, which showed that when significant radical sources are present in the non-chemical source terms, splitting the chemical and the transport sub-systems may incur O(1) errors unless the splitting time steps are comparable to or smaller than that required for explicit integration. As such, the transport term needs to be carried during the integration of stiff chemistry to avoid the large splitting errors. In AHI, fast species and reactions that may induce stiffness are treated implicitly, while the non-stiff variables and source terms, including slow reactions and the mixing term, are treated explicitly. The separation of fast-slow chemistry is performed on-the-fly based on analytically evaluated timescales for species and reactions, such that the complexity of the implicit core in the governing equations is minimized at each time step and the time-integration can be performed with high efficiency. The newly developed AHI2 scheme combines the midpoint scheme and the trapezoidal rule to achieve second-order accuracy. The second-order scheme is tested with a toy problem, as well as auto-ignition and unsteady perfectly stirred reactors (PSR) with detailed chemistry. Results show that AHI2 can significantly improve accuracy compared with AHI1. It was further found that AHI2 can accurately predict extinction of unsteady PSRs while the Strang splitting scheme fails to control the error, showing the necessity not to split the chemistry and transport source terms for prediction of extinction or forced-ignition problems involving significant radical sources. Further analysis of numerical efficiency shows that for auto-ignition AHI2 reduces computational cost primarily through the reduction in the number of variables to be solved implicitly, and the time-saving increases with the mechanism size, reaching approximately 70% for the 111-species USC-Mech II compared with a fully implicit scheme. For unsteady PSR involving homogeneous mixing, AHI2 achieved speedup factors of 20 to 30 compared with the Strang splitting scheme. Furthermore, sparse matrix techniques are integrated into AHI2 (AHI2-S) to achieve high computational efficiency. It is shown that the computational cost of AHI2-S is overall linearly proportional to the mechanism size and is comparable to that of evaluating reaction rates using CHEMKIN-II subroutines. It is further shown that AHI2-S achieves a speed-up factor of around two compared with the efficient fully implicit sparse solver LSODES with analytic Jacobian.