# Exponential integrators for the incompressible Navier-Stokes equations.

## Abstract

We provide an algorithm and analysis of a high order projection scheme for time integration of the incompressible Navier-Stokes equations (NSE). The method is based on a projection onto the subspace of divergence-free (incompressible) functions interleaved with a Krylov-based exponential time integration (KBEI). These time integration methods provide a high order accurate, stable approach with many of the advantages of explicit methods, and can reduce the computational resources over conventional methods. The method is scalable in the sense that the computational costs grow linearly with problem size. Exponential integrators, used typically to solve systems of ODEs, utilize matrix vector products of the exponential of the Jacobian on a vector. For large systems, this product can be approximated efficiently by Krylov subspace methods. However, in contrast to explicit methods, KBEIs are not restricted by the time step. While implicit methods require a solution of a linear system with the Jacobian, KBEIs only require matrix vector products of the Jacobian. Furthermore, these methods are based on linearization, so there is no non-linear system solve at each time step. Differential-algebraic equations (DAEs) are ordinary differential equations (ODEs) subject to algebraic constraints. The discretized NSE constitute a system of DAEs, where the incompressibility conditionmore »

- Authors:

- Publication Date:

- Research Org.:
- Sandia National Laboratories

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 975250

- Report Number(s):
- SAND2004-3742

TRN: US201009%%320

- DOE Contract Number:
- AC04-94AL85000

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; ALGORITHMS; BOUNDARY CONDITIONS; DIFFERENTIAL EQUATIONS; NAVIER-STOKES EQUATIONS; INCOMPRESSIBLE FLOW; Matrix theory; Navier stokes equations

### Citation Formats

```
Newman, Christopher K.
```*Exponential integrators for the incompressible Navier-Stokes equations.*. United States: N. p., 2004.
Web. doi:10.2172/975250.

```
Newman, Christopher K.
```*Exponential integrators for the incompressible Navier-Stokes equations.*. United States. doi:10.2172/975250.

```
Newman, Christopher K. Thu .
"Exponential integrators for the incompressible Navier-Stokes equations.". United States. doi:10.2172/975250. https://www.osti.gov/servlets/purl/975250.
```

```
@article{osti_975250,
```

title = {Exponential integrators for the incompressible Navier-Stokes equations.},

author = {Newman, Christopher K.},

abstractNote = {We provide an algorithm and analysis of a high order projection scheme for time integration of the incompressible Navier-Stokes equations (NSE). The method is based on a projection onto the subspace of divergence-free (incompressible) functions interleaved with a Krylov-based exponential time integration (KBEI). These time integration methods provide a high order accurate, stable approach with many of the advantages of explicit methods, and can reduce the computational resources over conventional methods. The method is scalable in the sense that the computational costs grow linearly with problem size. Exponential integrators, used typically to solve systems of ODEs, utilize matrix vector products of the exponential of the Jacobian on a vector. For large systems, this product can be approximated efficiently by Krylov subspace methods. However, in contrast to explicit methods, KBEIs are not restricted by the time step. While implicit methods require a solution of a linear system with the Jacobian, KBEIs only require matrix vector products of the Jacobian. Furthermore, these methods are based on linearization, so there is no non-linear system solve at each time step. Differential-algebraic equations (DAEs) are ordinary differential equations (ODEs) subject to algebraic constraints. The discretized NSE constitute a system of DAEs, where the incompressibility condition is the algebraic constraint. Exponential integrators can be extended to DAEs with linear constraints imposed via a projection onto the constraint manifold. This results in a projected ODE that is integrated by a KBEI. In this approach, the Krylov subspace satisfies the constraint, hence the solution at the advanced time step automatically satisfies the constraint as well. For the NSE, the projection onto the constraint is typically achieved by a projection induced by the L{sup 2} inner product. We examine this L{sup 2} projection and an H{sup 1} projection induced by the H{sup 1} semi-inner product. The H{sup 1} projection has an advantage over the L{sup 2} projection in that it retains tangential Dirichlet boundary conditions for the ow. Both the H{sup 1} and L{sup 2} projections are solutions to saddle point problems that are efficiently solved by a preconditioned Uzawa algorithm.},

doi = {10.2172/975250},

journal = {},

number = ,

volume = ,

place = {United States},

year = {2004},

month = {7}

}