Examining the accuracy of methods for obtaining pressure dependent rate coefficients.

The full energy-grained master equation (ME) is too large to be conveniently used in kinetic modeling, so almost always it is replaced by a reduced model using phenomenological rate coefficients. The accuracy of several methods for obtaining these pressure-dependent phenomenological rate coefficients, and so for constructing a reduced model, is tested against direct numerical solutions of the full ME, and the deviations are sometimes quite large. An algebraic expression for the error between the popular chemically-significant eigenvalue (CSE) method and the exact ME solution is derived. An alternative way to compute phenomenological rate coefficients, simulation least-squares (SLS), is presented. SLS is often about as accurate as CSE, and sometimes has significant advantages over CSE. One particular variant of SLS, using the matrix exponential, is as fast as CSE, and seems to be more robust. However, all of the existing methods for constructing reduced models to approximate the ME, including CSE and SLS, are inaccurate under some conditions, and sometimes they fail dramatically due to numerical problems. The challenge of constructing useful reduced models that more reliably emulate the full ME solution is discussed.