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

A weighted essentially non-oscillatory reconstruction scheme based on Hermite polynomials is developed and applied as a limiter for the discontinuous Galerkin 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 van Neumann neighborhood and are compact and consistent with the DG method. The developed HWENO 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 HWENO limiter for the DG methods.

[1]  Chi-Wang Shu,et al.  Runge-Kutta Discontinuous Galerkin Method Using WENO Limiters , 2005, SIAM J. Sci. Comput..

[2]  O. Friedrich,et al.  Weighted Essentially Non-Oscillatory Schemes for the Interpolation of Mean Values on Unstructured Grids , 1998 .

[3]  Timothy J. Barth,et al.  The design and application of upwind schemes on unstructured meshes , 1989 .

[4]  Jianxian Qiu,et al.  Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method II: Two dimensional case , 2005 .

[5]  Michael A. Leschziner,et al.  Average-State Jacobians and Implicit Methods for Compressible Viscous and Turbulent Flows , 1997 .

[6]  Rainald Löhner,et al.  A fast, matrix-free implicit method for compressible flows on unstructured grids , 1998 .

[7]  Jianxian Qiu,et al.  Weighted essential non‐oscillatory schemes for tidal bore on unstructured meshes , 2009 .

[8]  Thomas Sonar,et al.  On the construction of essentially non-oscillatory finite volume approximations to hyperbolic conservation laws on general triangulations : polynomial recovery, accuracy and stencil selection , 1997 .

[9]  Rémi Abgrall,et al.  On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation , 1994 .

[10]  Rainald Löhner,et al.  High-Reynolds number viscous flow computations using an unstructured-grid method , 2005 .

[11]  E. Toro,et al.  Restoration of the contact surface in the HLL-Riemann solver , 1994 .

[12]  Rainald Löhner,et al.  A p-multigrid discontinuous Galerkin method for the Euler equations on unstructured grids , 2006 .

[13]  Jianxian Qiu,et al.  Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: one-dimensional case , 2004 .

[14]  Rainald Löhner,et al.  On the computation of steady‐state compressible flows using a discontinuous Galerkin method , 2008 .

[15]  S. Osher,et al.  Uniformly high order accurate essentially non-oscillatory schemes, 111 , 1987 .

[16]  Chi-Wang Shu,et al.  The Runge-Kutta Discontinuous Galerkin Method for Conservation Laws V , 1998 .

[17]  Stanley Osher,et al.  UNIFORMLY HIGH ORDER ACCURATE , .

[18]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

[19]  George Em Karniadakis,et al.  The Development of Discontinuous Galerkin Methods , 2000 .

[20]  Rainald Löhner,et al.  A Fast p-Multigrid Discontinuous Galerkin Method for Compressible Flows at All Speeds , 2006 .

[21]  Chaowei Hu,et al.  No . 98-32 Weighted Essentially Non-Oscillatory Schemes on Triangular Meshes , 1998 .

[22]  A. Harten,et al.  Multi-Dimensional ENO Schemes for General Geometries , 1991 .