# An investigation of Newton-Krylov algorithms for solving incompressible and low Mach number compressible fluid flow and heat transfer problems using finite volume discretization

## Abstract

Fully coupled, Newton-Krylov algorithms are investigated for solving strongly coupled, nonlinear systems of partial differential equations arising in the field of computational fluid dynamics. Primitive variable forms of the steady incompressible and compressible Navier-Stokes and energy equations that describe the flow of a laminar Newtonian fluid in two-dimensions are specifically considered. Numerical solutions are obtained by first integrating over discrete finite volumes that compose the computational mesh. The resulting system of nonlinear algebraic equations are linearized using Newton`s method. Preconditioned Krylov subspace based iterative algorithms then solve these linear systems on each Newton iteration. Selected Krylov algorithms include the Arnoldi-based Generalized Minimal RESidual (GMRES) algorithm, and the Lanczos-based Conjugate Gradients Squared (CGS), Bi-CGSTAB, and Transpose-Free Quasi-Minimal Residual (TFQMR) algorithms. Both Incomplete Lower-Upper (ILU) factorization and domain-based additive and multiplicative Schwarz preconditioning strategies are studied. Numerical techniques such as mesh sequencing, adaptive damping, pseudo-transient relaxation, and parameter continuation are used to improve the solution efficiency, while algorithm implementation is simplified using a numerical Jacobian evaluation. The capabilities of standard Newton-Krylov algorithms are demonstrated via solutions to both incompressible and compressible flow problems. Incompressible flow problems include natural convection in an enclosed cavity, and mixed/forced convection past a backward facing step.

- Authors:

- Publication Date:

- Research Org.:
- EG and G Idaho, Inc., Idaho Falls, ID (United States)

- Sponsoring Org.:
- USDOE, Washington, DC (United States)

- OSTI Identifier:
- 130602

- Report Number(s):
- INEL-95/0118

ON: DE96002219; TRN: 95:008283

- DOE Contract Number:
- AC07-94ID13223

- Resource Type:
- Technical Report

- Resource Relation:
- Other Information: PBD: Oct 1995

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 42 ENGINEERING NOT INCLUDED IN OTHER CATEGORIES; 99 MATHEMATICS, COMPUTERS, INFORMATION SCIENCE, MANAGEMENT, LAW, MISCELLANEOUS; PARTIAL DIFFERENTIAL EQUATIONS; NUMERICAL SOLUTION; CONVERGENCE; COMPRESSIBLE FLOW; COMPUTERIZED SIMULATION; NAVIER-STOKES EQUATIONS; NONLINEAR PROBLEMS; PARALLEL PROCESSING; ALGORITHMS

### Citation Formats

```
McHugh, P R.
```*An investigation of Newton-Krylov algorithms for solving incompressible and low Mach number compressible fluid flow and heat transfer problems using finite volume discretization*. United States: N. p., 1995.
Web. doi:10.2172/130602.

```
McHugh, P R.
```*An investigation of Newton-Krylov algorithms for solving incompressible and low Mach number compressible fluid flow and heat transfer problems using finite volume discretization*. United States. https://doi.org/10.2172/130602

```
McHugh, P R. Sun .
"An investigation of Newton-Krylov algorithms for solving incompressible and low Mach number compressible fluid flow and heat transfer problems using finite volume discretization". United States. https://doi.org/10.2172/130602. https://www.osti.gov/servlets/purl/130602.
```

```
@article{osti_130602,
```

title = {An investigation of Newton-Krylov algorithms for solving incompressible and low Mach number compressible fluid flow and heat transfer problems using finite volume discretization},

author = {McHugh, P R},

abstractNote = {Fully coupled, Newton-Krylov algorithms are investigated for solving strongly coupled, nonlinear systems of partial differential equations arising in the field of computational fluid dynamics. Primitive variable forms of the steady incompressible and compressible Navier-Stokes and energy equations that describe the flow of a laminar Newtonian fluid in two-dimensions are specifically considered. Numerical solutions are obtained by first integrating over discrete finite volumes that compose the computational mesh. The resulting system of nonlinear algebraic equations are linearized using Newton`s method. Preconditioned Krylov subspace based iterative algorithms then solve these linear systems on each Newton iteration. Selected Krylov algorithms include the Arnoldi-based Generalized Minimal RESidual (GMRES) algorithm, and the Lanczos-based Conjugate Gradients Squared (CGS), Bi-CGSTAB, and Transpose-Free Quasi-Minimal Residual (TFQMR) algorithms. Both Incomplete Lower-Upper (ILU) factorization and domain-based additive and multiplicative Schwarz preconditioning strategies are studied. Numerical techniques such as mesh sequencing, adaptive damping, pseudo-transient relaxation, and parameter continuation are used to improve the solution efficiency, while algorithm implementation is simplified using a numerical Jacobian evaluation. The capabilities of standard Newton-Krylov algorithms are demonstrated via solutions to both incompressible and compressible flow problems. Incompressible flow problems include natural convection in an enclosed cavity, and mixed/forced convection past a backward facing step.},

doi = {10.2172/130602},

url = {https://www.osti.gov/biblio/130602},
journal = {},

number = ,

volume = ,

place = {United States},

year = {1995},

month = {10}

}