# Recovery Discontinuous Galerkin Jacobian-Free Newton-Krylov Method for All-Speed Flows

## Abstract

A novel numerical algorithm (rDG-JFNK) for all-speed fluid flows with heat conduction and viscosity is introduced. The rDG-JFNK combines the Discontinuous Galerkin spatial discretization with the implicit Runge-Kutta time integration under the Jacobian-free Newton-Krylov framework. We solve fully-compressible Navier-Stokes equations without operator-splitting of hyperbolic, diffusion and reaction terms, which enables fully-coupled high-order temporal discretization. The stability constraint is removed due to the L-stable Explicit, Singly Diagonal Implicit Runge-Kutta (ESDIRK) scheme. The governing equations are solved in the conservative form, which allows one to accurately compute shock dynamics, as well as low-speed flows. For spatial discretization, we develop a “recovery” family of DG, exhibiting nearly-spectral accuracy. To precondition the Krylov-based linear solver (GMRES), we developed an “Operator-Split”-(OS) Physics Based Preconditioner (PBP), in which we transform/simplify the fully-coupled system to a sequence of segregated scalar problems, each can be solved efficiently with Multigrid method. Each scalar problem is designed to target/cluster eigenvalues of the Jacobian matrix associated with a specific physics.

- Authors:

- Publication Date:

- Research Org.:
- Idaho National Laboratory (INL)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 936630

- Report Number(s):
- INL/CON-08-13822

TRN: US0805608

- DOE Contract Number:
- DE-AC07-99ID-13727

- Resource Type:
- Conference

- Resource Relation:
- Conference: ICCFD5,KOREA,07/07/2008,07/11/2008

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 22 GENERAL STUDIES OF NUCLEAR REACTORS; ACCURACY; ALGORITHMS; DIFFUSION; EIGENVALUES; FLUID FLOW; NAVIER-STOKES EQUATIONS; PHYSICS; SCALARS; STABILITY; VISCOSITY; Newton-Krylov; rDG-JFNK

### Citation Formats

```
Park, HyeongKae, Nourgaliev, Robert, Mousseau, Vincent, and Knoll, Dana.
```*Recovery Discontinuous Galerkin Jacobian-Free Newton-Krylov Method for All-Speed Flows*. United States: N. p., 2008.
Web.

```
Park, HyeongKae, Nourgaliev, Robert, Mousseau, Vincent, & Knoll, Dana.
```*Recovery Discontinuous Galerkin Jacobian-Free Newton-Krylov Method for All-Speed Flows*. United States.

```
Park, HyeongKae, Nourgaliev, Robert, Mousseau, Vincent, and Knoll, Dana. Tue .
"Recovery Discontinuous Galerkin Jacobian-Free Newton-Krylov Method for All-Speed Flows". United States. https://www.osti.gov/servlets/purl/936630.
```

```
@article{osti_936630,
```

title = {Recovery Discontinuous Galerkin Jacobian-Free Newton-Krylov Method for All-Speed Flows},

author = {Park, HyeongKae and Nourgaliev, Robert and Mousseau, Vincent and Knoll, Dana},

abstractNote = {A novel numerical algorithm (rDG-JFNK) for all-speed fluid flows with heat conduction and viscosity is introduced. The rDG-JFNK combines the Discontinuous Galerkin spatial discretization with the implicit Runge-Kutta time integration under the Jacobian-free Newton-Krylov framework. We solve fully-compressible Navier-Stokes equations without operator-splitting of hyperbolic, diffusion and reaction terms, which enables fully-coupled high-order temporal discretization. The stability constraint is removed due to the L-stable Explicit, Singly Diagonal Implicit Runge-Kutta (ESDIRK) scheme. The governing equations are solved in the conservative form, which allows one to accurately compute shock dynamics, as well as low-speed flows. For spatial discretization, we develop a “recovery” family of DG, exhibiting nearly-spectral accuracy. To precondition the Krylov-based linear solver (GMRES), we developed an “Operator-Split”-(OS) Physics Based Preconditioner (PBP), in which we transform/simplify the fully-coupled system to a sequence of segregated scalar problems, each can be solved efficiently with Multigrid method. Each scalar problem is designed to target/cluster eigenvalues of the Jacobian matrix associated with a specific physics.},

doi = {},

journal = {},

number = ,

volume = ,

place = {United States},

year = {2008},

month = {7}

}