# Using adaptive proper orthogonal decomposition to solve the reaction-diffusion equation

## Abstract

We introduce an adaptive POD method to reduce the computational cost of reacting flow simulations. The scheme is coupled with an operator-splitting algorithm to solve the reaction-diffusion equation. For the reaction sub-steps, locally valid basis vectors, obtained via POD and the method of snapshots, are used to project the minor species mass fractions onto a reduced dimensional space thereby decreasing the number of equations that govern combustion chemistry. The method is applied to a one-dimensional laminar premixed CH{sub 4}-air flame using GRImech 3.0; with errors less than 0:25%, a speed-up factor of 3:5 is observed. The speed-up results from fewer source term evaluations required to compute the Jacobian matrices.

- Authors:

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 946257

- Report Number(s):
- UCRL-JRNL-236975

TRN: US200903%%349

- DOE Contract Number:
- W-7405-ENG-48

- Resource Type:
- Journal Article

- Journal Name:
- Applied Numerical Mathematics, vol. 59, N/A, February 1, 2009, pp. 272-279

- Additional Journal Information:
- Journal Volume: 59

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 99 GENERAL AND MISCELLANEOUS; ALGORITHMS; CHEMISTRY; COMBUSTION; FLAMES; MATRICES; SOURCE TERMS; VECTORS

### Citation Formats

```
Singer, M A, and Green, W H.
```*Using adaptive proper orthogonal decomposition to solve the reaction-diffusion equation*. United States: N. p., 2007.
Web.

```
Singer, M A, & Green, W H.
```*Using adaptive proper orthogonal decomposition to solve the reaction-diffusion equation*. United States.

```
Singer, M A, and Green, W H. Mon .
"Using adaptive proper orthogonal decomposition to solve the reaction-diffusion equation". United States. https://www.osti.gov/servlets/purl/946257.
```

```
@article{osti_946257,
```

title = {Using adaptive proper orthogonal decomposition to solve the reaction-diffusion equation},

author = {Singer, M A and Green, W H},

abstractNote = {We introduce an adaptive POD method to reduce the computational cost of reacting flow simulations. The scheme is coupled with an operator-splitting algorithm to solve the reaction-diffusion equation. For the reaction sub-steps, locally valid basis vectors, obtained via POD and the method of snapshots, are used to project the minor species mass fractions onto a reduced dimensional space thereby decreasing the number of equations that govern combustion chemistry. The method is applied to a one-dimensional laminar premixed CH{sub 4}-air flame using GRImech 3.0; with errors less than 0:25%, a speed-up factor of 3:5 is observed. The speed-up results from fewer source term evaluations required to compute the Jacobian matrices.},

doi = {},

journal = {Applied Numerical Mathematics, vol. 59, N/A, February 1, 2009, pp. 272-279},

number = ,

volume = 59,

place = {United States},

year = {2007},

month = {12}

}

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