Fast approximations of exponential and logarithm functions combined with efficient storage/retrieval for combustion kinetics calculations

We developed two approaches to speed up combustion chemistry simulations by reducing the amount of time spent computing exponentials, logarithms, and complex temperature-dependent kinetics functions that heavily rely on them. The evaluation of these functions is very accurate in 64-bit arithmetic, but also slow. Since these functions span several orders of magnitude in temperature space, some of this accuracy can be traded with greater solution speed, provided that the governing ordinary differential equation (ODE) solver still grants user-defined solution convergence properties. The first approach tackled the exp() and log() functions, and replaced them with fast approximations which perform bit and integer operations on the exponential-based IEEE-754 floating point number machine representation. The second approach addresses complex temperature-dependent kinetics functions via storage/retrieval. We developed a function-independent piecewise polynomial approximation method with the following features: it minimizes table storage requirements, it is not subject to ill-conditioning over the whole variable range, it is of arbitrarily high order n > 0, and is fully vectorized. Formulations for both approaches are presented; and their performance assessed against zero-dimensional reactor simulations of hydrocarbon fuel ignition delay, with reaction mechanisms ranging from 10 to 104 species. The results show that, when used concurrently, both methods allow global speed-ups of about one order of magnitude even with an already highly-optimized sparse analytical Jacobian solver. The methods also demonstrate that global error is within the integrator's requested accuracy, and that the solver's performance is slightly positively affected, i.e., a slight reduction in the number of timesteps per integration is seen.