A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids.

*(English)*Zbl 1122.65089Summary: A weighted essentially non-oscillatory (WENO) reconstruction scheme based on Hermite polynomials is developed and applied as a limiter for the discontinuous Galerkin (DG) finite element method on unstructured grids. The solution polynomials are reconstructed using a WENO scheme by taking advantage of handily available and yet valuable information, namely the derivatives, in the context of the discontinuous Galerkin method. The stencils used in the reconstruction involve only the von Neumann neighborhood and are compact and consistent with the DG method.

The developed Hermite WENO limiter is implemented and used in a discontinuous Galerkin method to compute a variety of both steady-state and time-accurate compressible flow problems on unstructured grids. Numerical experiments for a wide range of flow conditions in both 2D and 3D configurations are presented to demonstrate the accuracy, effectiveness, and robustness of the designed Hermite WENO limiter for the DG methods.

The developed Hermite WENO limiter is implemented and used in a discontinuous Galerkin method to compute a variety of both steady-state and time-accurate compressible flow problems on unstructured grids. Numerical experiments for a wide range of flow conditions in both 2D and 3D configurations are presented to demonstrate the accuracy, effectiveness, and robustness of the designed Hermite WENO limiter for the DG methods.

##### MSC:

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

35L65 | Hyperbolic conservation laws |

76N15 | Gas dynamics, general |

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

65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |

##### Keywords:

compressible flows; unstructured grids; slope limiters; conservation laws; Euler equations; weighted essentially non-oscillatory (WENO) reconstruction scheme; Hermite polynomials; discontinuous Galerkin (DG) finite element method; numerical experiments
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\textit{H. Luo} et al., J. Comput. Phys. 225, No. 1, 686--713 (2007; Zbl 1122.65089)

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##### References:

[1] | Cockburn, B.; Shu, C.W., The runge – kutta discontinuous Galerkin method for conservation laws V: multidimensional system, Journal of computational physics, 141, 199-224, (1998) · Zbl 0920.65059 |

[2] | Cockburn, B.; Karniadakis, G.; Shu, C.W., The development of discontinuous Galerkin method, (), 5-50 |

[3] | H. Luo, J.D. Baum, R. Löhner, On the computation of steady-state compressible flows using a discontinuous Galerkin method, in: Proceedings of the Fourth International Conference on Computational Fluid Dynamics, Ghent, Belgium, 10-14, July, 2006. |

[4] | Harden, A.; Engquist, B.; Osher, S.; Chakravarthy, S.R., Uniformly high-order accurate essential non-oscillatory schemes III, Journal of computational physics, 71, 231-303, (1987) · Zbl 0652.65067 |

[5] | Liu, X.; Osher, S.; Chen, T.F., Weighted essential non-oscillatory schemes, Journal of computational physics, 115, 200-212, (1994) · Zbl 0811.65076 |

[6] | Qiu, J.; Shu, C.W., Runge – kutta discontinuous Galerkin method using WENO limiters, SIAM journal of scientific computing, 26, 907-929, (2005) · Zbl 1077.65109 |

[7] | Qiu, J.; Shu, C.W., Hermite WENO schemes and their application as limiters for runge – kutta discontinuous Galerkin method: one dimensional case, Journal of computational physics, 193, 115-135, (2004) · Zbl 1039.65068 |

[8] | Qiu, J.; Shu, C.W., Hermite WENO schemes and their application as limiters for runge – kutta discontinuous Galerkin method II: two dimensional case, Computers & fluids, 34, 642-663, (2005) · Zbl 1134.65358 |

[9] | A. Harden, S.R. Chakravarthy, Multidimensional ENO schemes on general geometries, ICASE Report No. 91-76, 1991. |

[10] | Abgrall, R., On essential non-oscillatory schemes on unstructured meshes, Journal of computational physics, 114, 45-58, (1994) · Zbl 0822.65062 |

[11] | Sonar, T., On the construction of essential non-oscillatory finite volume approximation to hyperbolic conservation laws on general triangulations: polynomial recovery, accuracy, and stencil selection, Computer methods in applied mechanics and engineering, 140, 157-182, (1997) · Zbl 0898.76086 |

[12] | Friedrich, O., Weighted essential non-oscillatory schemes for the interpolation of Mean values on unstructured grids, Journal of computational physics, 144, 194-212, (1998) · Zbl 1392.76048 |

[13] | Hu, C.; Shu, C.W., Weighted essential non-oscillatory schemes on unstructured triangular meshes, Journal of computational physics, 150, 97-127, (1999) · Zbl 0926.65090 |

[14] | Toro, E.F.; Spruce, M.; Speares, W., Restoration of the contact surface in the HLL-Riemann solver, Shock waves, 4, 25-34, (1994) · Zbl 0811.76053 |

[15] | Batten, P.; Leschziner, M.A.; Goldberg, U.C., Average-state Jacobians and implicit methods for compressible viscous and turbulent flows, Journal of computational physics, 137, 38-78, (1997) · Zbl 0901.76043 |

[16] | Luo, H.; Baum, J.D.; Löhner, R., High-Reynolds number viscous flow computations using an unstructured-grid method, Journal of aircraft, 42, 2, 483-492, (2005) |

[17] | Luo, H.; Baum, J.D.; Löhner, R., A p-multigrid discontinuous Galerkin method for the Euler equations on unstructured grids, Journal of computational physics, 211, 2, 767-783, (2006) · Zbl 1138.76408 |

[18] | H. Luo, J.D. Baum, R. Löhner, A fast, p-multigrid discontinuous Galerkin method for compressible flows at all speeds, AIAA Paper 2006-0110, 2006. |

[19] | Jiang, G.S.; Shu, C.W., Efficient implementation of weighted ENO schemes, Journal of computational physics, 126, 1, 202-228, (1996) · Zbl 0877.65065 |

[20] | Luo, H.; Baum, J.D.; Löhner, R.; Fast, A., Matrix-free implicit method for compressible flows on unstructured grids, Journal of computational physics, 146, 2, 664-690, (1998) · Zbl 0931.76045 |

[21] | T.J. Barth, D.C. Jespersen, The design of application of upwind schemes on unstructured grids, AIAA Paper 1989-0366, 1989. |

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