On the consistency of state vectors and Jacobian matrices

The formulation of reactive flow problems can be both quite challenging and important to the efficiency and robustness of solution algorithms. In this article, we focus on the choice of the thermochemical state vector as it relates to recently-developed computational techniques for complex combustion chemistry problems. We identify over-specification of the state vector as a source of both ambiguity and error in the partial derivatives used in forming analytical forms of the chemical source Jacobian matrix. We review and compare several approaches taken to increase sparsity of the Jacobian matrix, as it relates to the use of Newton-Krylov methods for implicit time integration, and identify proper techniques for achieving sparsity that do not rely on ad-hoc choice of state variables with inconsistent Jacobians. Chemical explosive mode analysis and linearly-implicit methods, such as Rosenbrock methods, are identified as areas where Jacobian accuracy may be critical. The distinction between how Jacobian exactness impacts Rosenbrock and Newton-Krylov methods is demonstrated with a simple example. We demonstrate the errors in conservation obtained from over-specification of the state vector with auto-ignition calculations for hydrogen, ethylene, and n-heptane chemistry with a high-order implicit Runge-Kutta method.