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**A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids.**
*(English)*
Zbl 1122.65089

Summary: 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 theory) |

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:

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