### Abstract

We report on our recent efforts on the formulation and the evaluation of a domain decomposition algorithm for the parallel solution of two-dimensional compressible inviscid flows. The starting point is a flow solver for the Euler equations, which is based on a mixed finite element/finite volume formulation on unstructured triangular meshes. Time integration of the resulting semi-discrete equations is obtained using a linearized backward Euler implicit scheme. As a result, each pseudo-time step requires the solution of a sparse linear system for the flow variables. In this study, a non-overlapping domain decomposition algorithm is used for advancing the solution at each implicit time step. First, we formulate an additive Schwarz algorithm using appropriate matching conditions at the subdomain interfaces. In accordance with the hyperbolic nature of the Euler equations, these transmission conditions are Dirichlet conditions for the characteristic variables corresponding to incoming waves. Then, we introduce interface operators that allow us to express the domain decomposition algorithm as a Richardson-type iteration on the interface unknowns. Algebraically speaking, the Schwarz algorithm is equivalent to a Jacobi iteration applied to a linear system whose matrix has a block structure. A substructuring technique can be applied to this matrix in order to obtain a fully implicit scheme in terms of interface unknowns. In our approach, the interface unknowns are numerical (normal) fluxes.

Language | English |
---|---|

Pages | 625-656 |

Number of pages | 32 |

Journal | International Journal for Numerical Methods in Fluids |

Volume | 37 |

Issue number | 6 |

Early online date | 24 Oct 2001 |

DOIs | |

Publication status | Published - 30 Nov 2001 |

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### Keywords

- domain decomposition method
- Euler equations
- finite elements
- finite volumes
- multigrid algorithm
- parallel computing
- triangular meshes

### Cite this

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*International Journal for Numerical Methods in Fluids*, vol. 37, no. 6, pp. 625-656. https://doi.org/10.1002/fld.184

**A domain decomposition approach to finite volume solutions of the Euler equations on unstructured triangular meshes.** / Dolean, Victoria; Lanteri, Stéphane.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A domain decomposition approach to finite volume solutions of the Euler equations on unstructured triangular meshes

AU - Dolean, Victoria

AU - Lanteri, Stéphane

PY - 2001/11/30

Y1 - 2001/11/30

N2 - We report on our recent efforts on the formulation and the evaluation of a domain decomposition algorithm for the parallel solution of two-dimensional compressible inviscid flows. The starting point is a flow solver for the Euler equations, which is based on a mixed finite element/finite volume formulation on unstructured triangular meshes. Time integration of the resulting semi-discrete equations is obtained using a linearized backward Euler implicit scheme. As a result, each pseudo-time step requires the solution of a sparse linear system for the flow variables. In this study, a non-overlapping domain decomposition algorithm is used for advancing the solution at each implicit time step. First, we formulate an additive Schwarz algorithm using appropriate matching conditions at the subdomain interfaces. In accordance with the hyperbolic nature of the Euler equations, these transmission conditions are Dirichlet conditions for the characteristic variables corresponding to incoming waves. Then, we introduce interface operators that allow us to express the domain decomposition algorithm as a Richardson-type iteration on the interface unknowns. Algebraically speaking, the Schwarz algorithm is equivalent to a Jacobi iteration applied to a linear system whose matrix has a block structure. A substructuring technique can be applied to this matrix in order to obtain a fully implicit scheme in terms of interface unknowns. In our approach, the interface unknowns are numerical (normal) fluxes.

AB - We report on our recent efforts on the formulation and the evaluation of a domain decomposition algorithm for the parallel solution of two-dimensional compressible inviscid flows. The starting point is a flow solver for the Euler equations, which is based on a mixed finite element/finite volume formulation on unstructured triangular meshes. Time integration of the resulting semi-discrete equations is obtained using a linearized backward Euler implicit scheme. As a result, each pseudo-time step requires the solution of a sparse linear system for the flow variables. In this study, a non-overlapping domain decomposition algorithm is used for advancing the solution at each implicit time step. First, we formulate an additive Schwarz algorithm using appropriate matching conditions at the subdomain interfaces. In accordance with the hyperbolic nature of the Euler equations, these transmission conditions are Dirichlet conditions for the characteristic variables corresponding to incoming waves. Then, we introduce interface operators that allow us to express the domain decomposition algorithm as a Richardson-type iteration on the interface unknowns. Algebraically speaking, the Schwarz algorithm is equivalent to a Jacobi iteration applied to a linear system whose matrix has a block structure. A substructuring technique can be applied to this matrix in order to obtain a fully implicit scheme in terms of interface unknowns. In our approach, the interface unknowns are numerical (normal) fluxes.

KW - domain decomposition method

KW - Euler equations

KW - finite elements

KW - finite volumes

KW - multigrid algorithm

KW - parallel computing

KW - triangular meshes

UR - http://www.scopus.com/inward/record.url?scp=0035976676&partnerID=8YFLogxK

UR - https://onlinelibrary.wiley.com/journal/10970363

U2 - 10.1002/fld.184

DO - 10.1002/fld.184

M3 - Article

VL - 37

SP - 625

EP - 656

JO - International Journal of Numerical Methods in Fluids

T2 - International Journal of Numerical Methods in Fluids

JF - International Journal of Numerical Methods in Fluids

SN - 0271-2091

IS - 6

ER -