New Finite Volume Weighted Essentially Nonoscillatory Schemes on Triangular Meshes

In this paper, we design a new type of high order finite volume weighted essentially nonoscillatory (WENO) schemes to solve hyperbolic conservation laws on triangular meshes. The main advantages of these schemes are their compactness and robustness and that they could maintain a good convergence property for some steady state problems. Compared with the classical finite volume WENO schemes [C. Hu and C.-W. Shu, J. Comput. Phys., 150 (1999), pp. 97--127], the optimal linear weights are independent of the topological structure of the triangular meshes and can be any positive numbers with the one requirement that their summation is one. This is the first time any high order accuracy with the usage of only five unequal sized stencils in a spatial reconstruction methodology on triangular meshes has been obtained. Extensive numerical results are provided to illustrate the good performance of such new finite volume WENO schemes.

[1]  Chi-Wang Shu,et al.  A technique of treating negative weights in WENO schemes , 2000 .

[2]  Michael Dumbser,et al.  Runge-Kutta Discontinuous Galerkin Method Using WENO Limiters , 2005, SIAM J. Sci. Comput..

[3]  Yuan Liu,et al.  A Robust Reconstruction for Unstructured WENO Schemes , 2013, J. Sci. Comput..

[4]  Jun Zhu,et al.  Runge-Kutta discontinuous Galerkin method using a new type of WENO limiters on unstructured meshes , 2013, J. Comput. Phys..

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

[6]  Dimitris Drikakis,et al.  WENO schemes for mixed-elementunstructured meshes , 2010 .

[7]  Michael Dumbser,et al.  Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems , 2007, J. Comput. Phys..

[8]  Harold L. Atkins,et al.  A Finite-Volume High-Order ENO Scheme for Two-Dimensional Hyperbolic Systems , 1993 .

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

[10]  S. Osher,et al.  Weighted essentially non-oscillatory schemes , 1994 .

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

[12]  Michael Dumbser,et al.  Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems , 2007, J. Comput. Phys..

[13]  P. Woodward,et al.  The numerical simulation of two-dimensional fluid flow with strong shocks , 1984 .

[14]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[15]  Jun Zhu,et al.  A New Type of Finite Volume WENO Schemes for Hyperbolic Conservation Laws , 2017, J. Sci. Comput..

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

[17]  Yong-Tao Zhang,et al.  Third Order WENO Scheme on Three Dimensional Tetrahedral Meshes , 2008 .

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

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

[20]  Jianxian Qiu,et al.  On the construction, comparison, and local characteristic decomposition for high-Order central WENO schemes , 2002 .

[21]  Chi-Wang Shu,et al.  A simple weighted essentially nonoscillatory limiter for Runge-Kutta discontinuous Galerkin methods , 2013, J. Comput. Phys..

[22]  Jun Zhu,et al.  A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws , 2016, J. Comput. Phys..

[23]  Ami Harten,et al.  Preliminary results on the extension of eno schemes to two-dimensional problems , 1987 .

[24]  Gabriella Puppo,et al.  Compact Central WENO Schemes for Multidimensional Conservation Laws , 1999, SIAM J. Sci. Comput..

[25]  G. Russo,et al.  Central WENO schemes for hyperbolic systems of conservation laws , 1999 .

[26]  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 .

[27]  Jun Zhu,et al.  A new third order finite volume weighted essentially non-oscillatory scheme on tetrahedral meshes , 2017, J. Comput. Phys..

[28]  Wai-Sun Don,et al.  An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws , 2008, J. Comput. Phys..

[29]  Jay Casper,et al.  Finite-volume implementation of high-order essentially nonoscillatory schemes in two dimensions , 1992 .

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

[31]  Wai-Sun Don,et al.  High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws , 2011, J. Comput. Phys..