Third-order Energy Stable WENO scheme

A new third-order Energy Stable Weighted Essentially Non-Oscillatory (ESWENO) finite difference scheme for scalar and vector hyperbolic equations with piecewise continuous initial conditions is developed. The new scheme is proven to be linearly stable in the energy norm for both continuous and discontinuous solutions. In contrast to the existing high-resolution shock-capturing schemes, no assumption that the reconstruction should be total variation bounded (TVB) is explicitly required to prove stability of the new scheme. We also present new weight functions which drastically improve the accuracy of the third-order ESWENO scheme. Based on a truncation error analysis, we show that the ESWENO scheme is design-order accurate for smooth solutions with any number of vanishing derivatives, if its tuning parameters satisfy certain constraints. Numerical results show that the new ESWENO scheme is stable and significantly outperforms the conventional third-order WENO scheme of Jiang and Shu in terms of accuracy, while providing essentially non-oscillatory solutions near strong discontinuities.

[1]  Shih-Hsien Yu,et al.  Discrete Shocks for Finite Difference Approximations to Scalar Conservation Laws , 1998 .

[2]  Rainald Löhner,et al.  A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids , 2007, J. Comput. Phys..

[3]  Chi-Wang Shu,et al.  Convergence of Godunov-Type Schemes for Scalar Conservation Laws under Large Time Steps , 2008, SIAM J. Numer. Anal..

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

[5]  Roberto Ferretti,et al.  Convergence of Semi-Lagrangian Approximations to Convex Hamilton-Jacobi Equations under (Very) Large Courant Numbers , 2002, SIAM J. Numer. Anal..

[6]  H. Kreiss,et al.  Finite Element and Finite Difference Methods for Hyperbolic Partial Differential Equations , 1974 .

[7]  Chi-Wang Shu,et al.  On a cell entropy inequality for discontinuous Galerkin methods , 1994 .

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

[9]  Michael B. Giles,et al.  Analysis of the accuracy of shock-capturing in the steady quasi-1D Euler equations , 1995 .

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

[11]  B. Strand.  Summation by parts for finite difference approximations for d/dx , 1994 .

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

[13]  Nail K. Yamaleev,et al.  A systematic methodology for constructing high-order energy stable WENO schemes , 2009, J. Comput. Phys..

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

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

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

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

[18]  Chi-Wang Shu,et al.  Monotonicity Preserving Weighted Essentially Non-oscillatory Schemes with Increasingly High Order of Accuracy , 2000 .

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

[20]  David Gottlieb,et al.  Spectral Methods on Arbitrary Grids , 1995 .

[21]  M. Carpenter,et al.  Accuracy of Shock Capturing in Two Spatial Dimensions , 1999 .

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

[23]  Chi-Wang Shu.  Total-variation-diminishing time discretizations , 1988 .

[24]  D. Gottlieb,et al.  Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes , 1994 .

[25]  Zhi J. Wang,et al.  Optimized weighted essentially nonoscillatory schemes for linear waves with discontinuity: 381 , 2001 .

[26]  P. Lax,et al.  Systems of conservation laws , 1960 .

[27]  S. Osher,et al.  On the convergence of difference approximations to scalar conservation laws , 1988 .

[28]  Magnus Svärd,et al.  Stable and Accurate Artificial Dissipation , 2004, J. Sci. Comput..

[29]  D. Gottlieb,et al.  A Stable and Conservative Interface Treatment of Arbitrary Spatial Accuracy , 1999 .

[30]  J. M. Powers,et al.  Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points , 2005 .

[31]  Huanan Yang,et al.  On Wavewise Entropy Inequality for High Resolution Schemes II: Fully Discrete MUSCL Schemes with Exact Evolution in Small Time , 1998 .

[32]  V. Gregory Weirs,et al.  A bandwidth-optimized WENO scheme for the effective direct numerical simulation of compressible turbulence , 2006, J. Comput. Phys..

[33]  J. Hesthaven,et al.  Nodal high-order methods on unstructured grids , 2002 .

[34]  Lakshmi N. Sankar,et al.  Uniformly High-Order Essentially Nonoscillatory Schemes for Vortex Convection Across Overset Interfaces , 2008 .

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

[36]  Jan S. Hesthaven,et al.  A Stable Penalty Method for the Compressible Navier-Stokes Equations: I. Open Boundary Conditions , 1996, SIAM J. Sci. Comput..

[37]  Songming Hou,et al.  Solutions of Multi-dimensional Hyperbolic Systems of Conservation Laws by Square Entropy Condition Satisfying Discontinuous Galerkin Method , 2007, J. Sci. Comput..