# A semi-implicit numerical scheme for reacting flow. II. Stiff, operator-split formulation

## Abstract

A stiff, 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. 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 ordermore »

- Authors:

- Publication Date:

- Research Org.:
- Johns Hopkins Univ., Baltimore, MD (US)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 20000636

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 154; Journal Issue: 2; Other Information: PBD: 20 Sep 1999; Journal ID: ISSN 0021-9991

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 40 CHEMISTRY; 99 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; COMBUSTION KINETICS; NUMERICAL SOLUTION; UNSTEADY FLOW; TWO-DIMENSIONAL CALCULATIONS; METHANE; FLAME PROPAGATION

### Citation Formats

```
Knio, O M, Najm, H N, and Wyckoff, P S.
```*A semi-implicit numerical scheme for reacting flow. II. Stiff, operator-split formulation*. United States: N. p., 1999.
Web. doi:10.1006/jcph.1999.6322.

```
Knio, O M, Najm, H N, & Wyckoff, P S.
```*A semi-implicit numerical scheme for reacting flow. II. Stiff, operator-split formulation*. United States. https://doi.org/10.1006/jcph.1999.6322

```
Knio, O M, Najm, H N, and Wyckoff, P S. Mon .
"A semi-implicit numerical scheme for reacting flow. II. Stiff, operator-split formulation". United States. https://doi.org/10.1006/jcph.1999.6322.
```

```
@article{osti_20000636,
```

title = {A semi-implicit numerical scheme for reacting flow. II. Stiff, operator-split formulation},

author = {Knio, O M and Najm, H N and Wyckoff, P S},

abstractNote = {A stiff, 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. 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 C{sub 1}C{sub 2} 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.},

doi = {10.1006/jcph.1999.6322},

url = {https://www.osti.gov/biblio/20000636},
journal = {Journal of Computational Physics},

issn = {0021-9991},

number = 2,

volume = 154,

place = {United States},

year = {1999},

month = {9}

}