A Semi-implicit Numerical Scheme for Reacting Flow: II. Stiff, Operator-Split Formulation

Omar M. Knio*, Habib N. Najm, Peter S. Wyckoff

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

153 Scopus citations


A stiff,1 operator-split projection scheme is constructed for the simulation of unsteady two-dimensional reacting flow with detailed kinetics. The scheme is based on the compressible conservation equations for an ideal gas mixture in the zero-Mach-number limit. The equations of motion are spatially discretized using second-order centered differences and are advanced in time using a new stiff predictor-corrector approach. The new scheme is a modified version of the additive, stiff scheme introduced in a previous effort by H. N. Najm, P. S. Wyckoff, and O. M. Knio (1998, J. Comput. Phys. 143, 381). The predictor updates the scalar fields using a Strang-type operator-split integration step which combines several explicit diffusion sub-steps with a single stiff step for the reaction terms, such that the global time step may significantly exceed the critical diffusion stability limit. Convection terms are explicitly handled using a second-order multi-step scheme. The velocity field is advanced using a projection scheme which consists of a partial convection-diffusion update followed by a pressure correction step. A split approach is also adopted for the convection-diffusion step in the momentum update. This splitting combines an explicit treatment of the convective terms at the global time step with several explicit fractional steps for diffusion. Finally, a corrector step is implemented in order to couple the evolution of the density and velocity fields and to stabilize the computations. The corrector acts only on the convective terms and the pressure field, while the predicted updates due to diffusion and reaction are left unchanged. The correction of the scalar fields is implemented using a single-step non-split, non-stiff, second-order time integration. A similar procedure is used for the velocity field, which is followed by a pressure projection step. The performance and behavior of the operator-split scheme are first analyzed based on tests for a nonlinear reaction-diffusion equation in one space dimension, followed by computations with a detailed C1C2 methane-air mechanism in one and two dimensions. The tests are used to verify that the scheme is effectively second order in time, and to suggest guidelines for selecting integration parameters, including the number of fractional diffusion steps and tolerance levels in the stiff integration. For two-dimensional simulations with the present reaction mechanism, flame conditions, and resolution parameters, speedup factors of about 5 are achieved over the previous additive scheme, and about 25 over the original explicit scheme.

Original languageEnglish (US)
Pages (from-to)428-467
Number of pages40
JournalJournal of Computational Physics
Issue number2
StatePublished - Sep 20 1999
Externally publishedYes


  • Chemistry
  • Flow
  • Implicit
  • Operator splitting
  • Projection
  • Reacting
  • Stiff

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • General Physics and Astronomy
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


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