Linearly Implicit Integration of Vibrational Master Equation Using Automatic Differentiation

Vibrational nonequilibrium of diatomic molecules resulting in non-Boltzmann distributions is a phenomenon commonly encountered in rapid hypersonic nozzle expansions. The usual assumption of different vibrational temperatures for different species cannot simulate such nonequilibrium behaviour. State-to-state models governed by a master equation describe the evolution of population number for all vibrational levels. These equations possess singularly perturbed characteristics and stable numerical integration must account for this stiffness. Linearly implicit, or Rosenbrock-Wanner (ROW), integration methods are inherently more stable than explicit methods and less computationally complex than implicit ones of equivalent order. One historical limitation of ROW methods is the requirement that the exact Jacobian of the evolution rate vector must be computed to achieve classical order of convergence. This may be analytically intractable for large systems. Automatic Differentiation (AD) provides a solution by algorithmically determining the exact derivatives, to working precision, based upon implementations of the rate vector. Stateto-state simulations of carbon monoxide vibrational kinetics under the influence of sudden a reduction in transitional/rotational temperature using linearly implicit time integration with AD is presented. Computational performance of the ROW method is compared to second-order backward-difference integration.