A moving mesh WENO method based on exponential polynomials for one-dimensional conservation laws

Abstract In the article [Yang et al. (2012) [37] ], the authors have developed a high order moving mesh WENO method for one-dimensional (1D) hyperbolic conservation laws, which is shown to be effective in resolving shocks and other complex solution structures. In this paper, in the light of the similar moving mesh technique, we develop a novel WENO scheme with non-polynomial bases, in particular, the exponential bases to further improve the performance of WENO schemes for solving 1D conservation laws. Furthermore, we modify the original moving mesh technique by developing a new monitor function as well as a different mesh smoothing strategy. A collection of numerical examples is presented to demonstrate high order accuracy and robustness of the method in capturing smooth and non-smooth solutions including the strong δ shock arising from the weakly hyperbolic pressureless Euler equations.

[1]  Jungho Yoon,et al.  Sixth-order Weighted Essentially Nonoscillatory Schemes Based on Exponential Polynomials , 2016, SIAM J. Sci. Comput..

[2]  Weizhang Huang,et al.  Moving Mesh Methods Based on Moving Mesh Partial Differential Equations , 1994 .

[3]  Tao Tang,et al.  Adaptive Mesh Methods for One- and Two-Dimensional Hyperbolic Conservation Laws , 2003, SIAM J. Numer. Anal..

[4]  Robert D. Russell,et al.  Adaptive Moving Mesh Methods , 2010 .

[5]  Bruno Després,et al.  A cell-centered Lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension , 2009, J. Comput. Phys..

[6]  Jianxian Qiu,et al.  Trigonometric WENO Schemes for Hyperbolic Conservation Laws and Highly Oscillatory Problems , 2010 .

[7]  Oliver Kolb,et al.  On the Full and Global Accuracy of a Compact Third Order WENO Scheme , 2014, SIAM J. Numer. Anal..

[8]  John M. Stockie,et al.  A Moving Mesh Method for One-dimensional Hyperbolic Conservation Laws , 2000, SIAM J. Sci. Comput..

[9]  Xiaobo Yang,et al.  A Moving Mesh WENO Method for One-Dimensional Conservation Laws , 2012, SIAM J. Sci. Comput..

[10]  Jungho Yoon,et al.  A short note on the error estimates of Yuan-Shu discontinuous Galerkin method based on non-polynomial approximation spaces , 2016, J. Comput. Phys..

[11]  Michael Dumbser,et al.  On Arbitrary-Lagrangian-Eulerian One-Step WENO Schemes for Stiff Hyperbolic Balance Laws , 2012, 1207.6407.

[12]  Robert D. Russell,et al.  Moving Mesh Strategy Based on a Gradient Flow Equation for Two-Dimensional Problems , 1998, SIAM J. Sci. Comput..

[13]  Weizhang Huang,et al.  A high dimensional moving mesh strategy , 1998 .

[14]  Bruno Després,et al.  Lagrangian Gas Dynamics in Two Dimensions and Lagrangian systems , 2005 .

[15]  Pierre-Henri Maire,et al.  A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes , 2009, J. Comput. Phys..

[16]  Chi-Wang Shu,et al.  Discontinuous Galerkin method for Krause's consensus models and pressureless Euler equations , 2013, J. Comput. Phys..

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

[18]  Chaopeng Shen,et al.  Adaptive mesh refinement based on high order finite difference WENO scheme for multi-scale simulations , 2011, J. Comput. Phys..

[19]  W. J. Studden,et al.  Tchebycheff Systems: With Applications in Analysis and Statistics. , 1967 .

[20]  Michael Dumbser,et al.  A direct Arbitrary-Lagrangian-Eulerian ADER-WENO finite volume scheme on unstructured tetrahedral meshes for conservative and non-conservative hyperbolic systems in 3D , 2014, J. Comput. Phys..

[21]  Yan Jiang,et al.  An Alternative Formulation of Finite Difference Weighted ENO Schemes with Lax-Wendroff Time Discretization for Conservation Laws , 2013, SIAM J. Sci. Comput..

[22]  Pingwen Zhang,et al.  An adaptive mesh redistribution method for nonlinear Hamilton--Jacobi equations in two-and three-dimensions , 2003 .

[23]  Zhijun Tan,et al.  Moving mesh methods for Boussinesq equation , 2009 .

[24]  Michael Dumbser,et al.  Arbitrary-Lagrangian–Eulerian ADER–WENO finite volume schemes with time-accurate local time stepping for hyperbolic conservation laws , 2014, 1402.6897.

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

[26]  Yan Jiang,et al.  Free-stream preserving finite difference schemes on curvilinear meshes , 2014 .

[27]  Weizhang Huang,et al.  Moving mesh partial differential equations (MMPDES) based on the equidistribution principle , 1994 .

[28]  Chi-Wang Shu.  Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws , 1998 .

[29]  Chi-Wang Shu,et al.  High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems , 2009, SIAM Rev..

[30]  Rémi Abgrall,et al.  Cell-centered discontinuous Galerkin discretizations for two-dimensional scalar conservation laws on unstructured grids and for one-dimensional Lagrangian hydrodynamics , 2011 .

[31]  Chi-Wang Shu,et al.  A high order ENO conservative Lagrangian type scheme for the compressible Euler equations , 2007, J. Comput. Phys..

[32]  Claus-Dieter Munz,et al.  On Godunov-type schemes for Lagrangian gas dynamics , 1994 .

[33]  Hongjoon Kim,et al.  ADAPTIVE MESH REFINEMENT FOR WEIGHTED ESSENTIALLY NON-OSCILLATORY SCHEMES , 2008 .

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

[35]  P. Lax.  Weak solutions of nonlinear hyperbolic equations and their numerical computation , 1954 .

[36]  Rémi Abgrall,et al.  A discontinuous Galerkin discretization for solving the two-dimensional gas dynamics equations written under total Lagrangian formulation on general unstructured grids , 2014, J. Comput. Phys..

[37]  Rémi Abgrall,et al.  A Cell-Centered Lagrangian Scheme for Two-Dimensional Compressible Flow Problems , 2007, SIAM J. Sci. Comput..

[38]  Jun Zhu,et al.  WENO Schemes and Their Application as Limiters for RKDG Methods Based on Trigonometric Approximation Spaces , 2013, J. Sci. Comput..