##
**High-order accurate discontinuous finite element solution of the 2D Euler equations.**
*(English)*
Zbl 0902.76056

Summary: This paper deals with a high-order accurate discontinuous finite element method for the numerical solution of the Euler equations. The method combines two key ideas, the basis of the finite volume and of the finite element method, the physics of wave propagation being accounted for by means Riemann problems and accuracy being obtained by means of high-order polynomial approximations within elements. We focus our attention on two-dimensional steady-state problems and present higher order accurate (up to fourth-order) discontinuous finite element solutions on unstructured grids of triangles. In particular, we show that, in the presence of curved boundaries, a meaningful high-order accurate solution can be obtained only if a corresponding high-order approximation of the geometry is employed. We present numerical solutions of classical test cases computed with linear, quadratic, and cubic elements which illustrate the versatility of the method and the importance of the boundary condition treatment.

### MSC:

76M10 | Finite element methods applied to problems in fluid mechanics |

76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |

### Keywords:

wave propagation; Riemann problems; high-order polynomial approximations; unstructured grids of triangles; curved boundaries
PDF
BibTeX
XML
Cite

\textit{F. Bassi} and \textit{S. Rebay}, J. Comput. Phys. 138, No. 2, 251--285 (1997; Zbl 0902.76056)

Full Text:
DOI

### References:

[1] | AGARD, AR-211 (1985) |

[2] | Barth, T. J.; Frederickson, P. O., Higher Order Solution of the Euler Equations on Unstructured Grids Using Quadratic Reconstruction, AIAA, 90-0013 (1990) |

[3] | Bassi, F.; Rebay, S.; Savini, M., A high resolution discontinuous Galerkin method for hyperbolic problems on unstructured grids, III-rd ICFD Conference, Reading, April 1992. III-rd ICFD Conference, Reading, April 1992, Numerical Methods in Fluid Dynamics (1993), Clarendon Press: Clarendon Press Oxford, p. 345- · Zbl 0808.76043 |

[4] | Bassi, F.; Rebay, S.; Savini, M., Discontinuous finite element Euler solutions on unstructured adaptive grids, (Napolitano, M.; Sabetta, F., XIIIth ICNMFD, Rome, July 6-10, 1992. XIIIth ICNMFD, Rome, July 6-10, 1992, Lecture Notes in Physics, 414 (1993), Springer-Verlag: Springer-Verlag New York/Berlin), 245 |

[5] | Bassi, F.; Rebay, S., Accurate 2D Euler computations by means of a high order discontinuous finite element method, XIVth ICNMFD, Bangalore, July 11-15, 1994. XIVth ICNMFD, Bangalore, July 11-15, 1994, Lecture Notes in Physics (1996), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0850.76344 |

[6] | Bassi, F.; Rebay, S., Discontinuous finite element high order accurate numerical solution of the compressible Navier-Stokes equations, (Baines, M. J.; Morton, K. W., IVth ICFD Conference, Oxford, April 3-6, 1995. IVth ICFD Conference, Oxford, April 3-6, 1995, Numerical Methods in Fluid Dynamics (1995), Clarendon Press: Clarendon Press Oxford) · Zbl 0923.76110 |

[7] | Bey, K. S.; Oden, J. T., A Runge-Kutta Discontinuous Finite Element Method for High Speed Flows, AIAA Paper, 91-1575-CP, 541 (1991) |

[8] | Biswas, R.; Devine, K. D.; Flaherty, J. E., Parallel, adaptive finite element methods for conservation laws, Appl. Numer. Math., 14, 255 (1994) · Zbl 0826.65084 |

[9] | Cockburn, B.; Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework, Math. Comput., 52, 411 (1989) · Zbl 0662.65083 |

[10] | Cockburn, B.; Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One dimensional systems, J. Comput. Phys., 84, 90 (1989) · Zbl 0677.65093 |

[11] | Cockburn, B.; Hou, S.; Shu, C.-W., The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case, Math. Comput., 54, 545 (1990) · Zbl 0695.65066 |

[12] | Cockburn, B.; Shu, C.-W., The \(P^1 RKDG \), ICASE Report, 91-32 (1991) |

[13] | Dadone, A.; Grossman, B., Surface Boundary Conditions for the Numerical Solution of the Euler Equations, AIAA Journal, 32, 285-293 (1994) · Zbl 0800.76323 |

[14] | Godunov, S. K., A Difference Scheme for Numerical Computation of Discontinuous Solutions of Hydrodynamic Equations, Math Sb., 47, 271 (1959) · Zbl 0171.46204 |

[15] | Harten, A.; Chakravarthy, S. R., Multi-Dimensional ENO Schemes for General Geometries, ICASE Report, 91-76 (1991) |

[17] | Lesaint, P.; Raviart, P. A., On a finite element method to solve the neutron transport equation, (de Boor, C., Mathematical Aspects of Finite Elements in Partial Differential Equations (1974), Academic Press: Academic Press New York), 89 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.