Explicit Time-Scale Splitting Algorithm for Stiff Problems: Auto-ignition of Gaseous Mixtures behind a Steady Shock

A new explicit algorithm based on the computational singular perturbation (CSP) method is presented. This algorithm is specifically designed to solve stiff problems, and its performance increases with stiffness. The key concept in its structure is the splitting of the fast from the slow time scales in the problem, realized by embedding CSP concepts into an explicit scheme. In simple terms, the algorithm marches in time with only the terms producing the slow time scales, while the contribution of the terms producing the fast time scales is taken into account at the end of each integration step as a correction. The new algorithm is designed for the integration of stiff systems of PDEs by means of explicit schemes. For simplicity in the presentation and discussion of the different features of the new algorithm, a simple test case is considered, involving the auto-ignition of a methane/air mixture behind a normal shock wave, which is described by a system of ODEs. The performance of the new algorithm (accuracy and computational efficiency) is then compared with the well-known LSODE package. Its merits when used for the solution of systems of PDEs are discussed. Although when dealing with a stiff system of ODEs the new algorithm is shown to provide equal accuracy with that delivered by LSODE at the cost of higher execution time, the results indicate that its performance could be superior when facing a stiff system of PDEs.