On the merits of extrapolation-based stiff ODE solvers for combustion CFD

In applications including compression-ignition engines, there is a need to accommodate more realistic chemistry in computational fluid dynamics (CFD) simulations. Here, we consider approaches where a chemical mechanism is implemented in an application CFD code, an operator-splitting strategy is used to isolate the chemical source terms, and a stiff ordinary differential equation (ODE) solver is used to compute the changes in composition due to chemical reactions for each computational element. Chemical source terms often dominate the computational effort, and reducing the high computational cost associated with realistic chemistry has been the subject of extensive research. This includes work on improved stiff ODE solvers, which in most cases has centered on backward differentiation formula (BDF) methods. Here a different class of solvers is considered, based on extrapolation methods. Key elements of stiff ODE solvers are reviewed briefly, focusing on differences between BDF methods and extrapolation methods. Issues related to using a stiff ODE solver with operator splitting in a CFD code (where the solver is repeatedly stopped and restarted) are emphasized. Homogeneous-reactor results are presented first. There the relationship between user-specified error tolerances and solution accuracy is explored, tradeoffs between accuracy and CPU time are shown, and close-to-linear increase in CPU time with increasing chemical mechanism size is demonstrated. Engine results are presented next, including both homogeneous-charge compression-ignition engines, and direct-injection (nonhomogeneous) compression-ignition engines. There some results are presented where the stiff ODE solver is combined with a dynamic adaptive chemistry scheme. In all cases, it is found that the extrapolation solver offers significant advantages in accuracy and computational efficiency compared to the BDF solver. While the results presented are for one BDF solver (CVODE) and one extrapolation solver (SEULEX), it is anticipated that the insight into how stiff ODE solvers behave in combustion CFD will be broadly applicable to other solvers in these general classes.