A smoothness indicator constant for sine functions

Abstract Smoothness indicator is an essential part of weighted essentially non-oscillatory (WENO) scheme, whose target is to distinguish discontinuous profiles and continuous profiles. However, the magnitudes of most of the smoothness indicators will be decreased significantly when the stencil approaches critical points, which would be negative for the numerical simulation. The reason should be that the first derivative term which will drop to almost zero occupies a large proportion in these smoothness indicators. To decrease the variances on different stencils in smooth region, smoothness indicators could be required to be constant for a specific kind of smooth functions, and the best choice should be the sine functions. In the present paper, a new smoothness indicator on four-point stencil is constructed based on this criterion. Compared with the classical smoothness indicator (Jiang and Shu, 1996, [3] ), the new one has a more succinct form, and takes less floating point operations which means that it will be more time efficient in the computation. Furthermore, the new smoothness indicator will get a larger variation as the sub-stencil moves from a smooth profile to a discontinuous profile, which would be helpful for stability near discontinuities. By using the new smoothness indicator, a seven-point WENO scheme is constructed which will reduce to the underlying linear scheme for monochromatic waves. As a result, it behaves the same as the underlying linear scheme for approximate dispersion relation (ADR). According to properties of the proposed smoothness indicator, this scheme should have excellent performance for profiles close to sine waves, better stability near discontinuities, and higher time efficiency. Numerical simulations predict the good performance of the proposed scheme.

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

[2]  Erik Dick,et al.  On the spectral and conservation properties of nonlinear discretization operators , 2011, J. Comput. Phys..

[3]  Antonio Baeza,et al.  On the Efficient Computation of Smoothness Indicators for a Class of WENO Reconstructions , 2019, Journal of Scientific Computing.

[4]  Bruno Costa,et al.  An improved WENO-Z scheme , 2016, J. Comput. Phys..

[5]  Wai-Sun Don,et al.  Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes , 2013, J. Comput. Phys..

[6]  Lei Luo,et al.  A sixth order hybrid finite difference scheme based on the minimized dispersion and controllable dissipation technique , 2014, J. Comput. Phys..

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

[8]  Yuxin Ren,et al.  A characteristic-wise hybrid compact-WENO scheme for solving hyperbolic conservation laws , 2003 .

[9]  Sergio Pirozzoli,et al.  On the spectral properties of shock-capturing schemes , 2006, J. Comput. Phys..

[10]  Mark H. Carpenter,et al.  Computational Considerations for the Simulation of Shock-Induced Sound , 1998, SIAM J. Sci. Comput..

[11]  G. A. Gerolymos,et al.  Very-high-order weno schemes , 2009, J. Comput. Phys..

[12]  W. Don,et al.  High order Hybrid central-WENO finite difference scheme for conservation laws , 2007 .

[13]  Chi-Wang Shu,et al.  A New Smoothness Indicator for the WENO Schemes and Its Effect on the Convergence to Steady State Solutions , 2007, J. Sci. Comput..

[14]  Nikolaus A. Adams,et al.  An adaptive central-upwind weighted essentially non-oscillatory scheme , 2010, J. Comput. Phys..

[15]  Yeon Ju Lee,et al.  An improved weighted essentially non-oscillatory scheme with a new smoothness indicator , 2013, J. Comput. Phys..

[16]  Sergio Pirozzoli,et al.  A general framework for the evaluation of shock-capturing schemes , 2019, J. Comput. Phys..

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

[18]  David P. Lockard,et al.  High-accuracy algorithms for computational aeroacoustics , 1995 .

[19]  Gecheng Zha,et al.  Improvement of the WENO scheme smoothness estimator , 2008 .

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

[21]  C. Tam,et al.  Dispersion-relation-preserving finite difference schemes for computational acoustics , 1993 .

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

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

[24]  Naga Raju Gande,et al.  Third‐order WENO scheme with a new smoothness indicator , 2017 .

[25]  Jianhan Liang,et al.  A new smoothness indicator for third‐order WENO scheme , 2016 .

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

[27]  Christopher K. W. Tam,et al.  Benchmark problems and solutions , 1995 .

[28]  Nikolaus A. Adams,et al.  A family of high-order targeted ENO schemes for compressible-fluid simulations , 2016, J. Comput. Phys..

[29]  Ping Fan,et al.  High order weighted essentially nonoscillatory WENO-η schemes for hyperbolic conservation laws , 2014, J. Comput. Phys..

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

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

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

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