Computational singular perturbation is a useful technique for the analysis, reduction and time integration of stiff ordinary differential equations systems. This happens when the required numerical integration step size of the ordinary differential equation system is very minute when compared with the time domain of concern. Luckily, this system is designed to deal such intricate and stiff kinetics problems using a refinement procedure that is relevant to Eigen computation. Unfortunately, the high computational costs of the Eigen computation make it difficult to apply computational singular perturbation on-the-fly in practical combustion simulations, despite its it capability to systematically identify exhausted fast modes and approximate the associated ordinary differential equations with algebraic equations to remove the chemical stiffness.
In a recent paper published in the scientific journal, Combustion science and technology, professor Sau-Hai Lam at Princeton University demonstrated the possibility of efficient catalytic or semi-analytic approaches in enabling on-the-fly chemical stiffness removal using computational singular perturbation. In his work, he clarified novel computational singular perturbation formulation through numerical computations.
Professor Sau-Hai Lam began his analysis by presenting a demonstration of the problem where he employed a model having three unknowns in the formulation of the chemical kinetics. He then chose a sufficiently small time step so as to ensure the mathematical correctness of the computed solutions. To study the reaction system, the professor computed the current numerical value of τx (x, y, z) for many values of x, the numbers were the most valued computational singular perturbation diagnostic data. He then considered alternative representations which enabled the assumption of τ not being dependent on x and replacement of the initial ordinary differential equations with the algebraic equation.
The author observed that the novel computational singular perturbation approach could deal the interesting couplings between reduced chemistry and diffusion, for the reacting flow terms, in instances where the governing partial differential equations included diffusion terms. Most importantly, he noted that for the new computational singular perturbation approach, no exorbitant (cost wise) Eigen-calculations were required or involved.
The study has successfully presented an efficient formulation of computational singular perturbation for toy problems consisting of three variables. It has been demonstrated that linear combinations of fast variables may result in slow modes. Moreover, the new computational singular perturbation technique has provided the mathematical rationale on how to select and update the current acceptable time step when doing numerical integration on stiff ordinary differential equations systems.
In conclusion, the presented approach can and should be exploited when the goal is to efficiently derive reduced models or to numerically integrate stiff ordinary differential equations systems. To this regard, analytic or semi-analytic approaches avoiding the expensive Eigen-decomposition should be exploited.
S. H. Lam. An Efficient Implementation of Computational Singular Perturbation. Combustion science and technology 2018, vol. 190, no. 1, 157–163
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