The Artificial Neuron Indicator in Hybrid Weighted Essentially Non-oscillatory Schemes

Abstract

In this paper, we propose a troubled-cell indicator in hybrid weighted essentially non-oscillatory (WENO) schemes by training an artificial neuron (AN), which is denoted as the AN indicator. The key idea of hybrid WENO schemes is to use discontinuity detectors to identify non-smooth regions, applying the costly WENO approximation in discontinuous regions and the cheaper linear approximation in smooth regions. The AN indicator has the advantages of being efficient, low dissipation, and high resolution. To analyze the performance of the AN indicator, we compare it with the multi-resolution (MR) indicator on one-dimensional (1D) uniform grids and the modified multi-resolution (MMR) indicator on two-dimensional (2D) unstructured triangular meshes for hyperbolic conservation laws. Numerical experiments show that, compared to the MR or MMR indicator, the AN indicator has a smaller percentage of reconstruction by the WENO approximation and more accurate numerical results. Moreover, these indicators significantly reduce computational costs.

Keywords

Hyperbolic conservation laws / Hybrid weighted essentially nonoscillatory (WENO) scheme / Artificial neuron (AN) / Troubled-cell indicator / 35L65 / 65M60

References

[1] Cockburn B, Shu C-W. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Math. Comp., 1989, 52(186): 411-435

[2] Cockburn B, Shu C-W. The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys., 1998, 141(2): 199-224

[3] Cockburn B, Shu C-W. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput., 2001, 16: 173-261

[4] Cockburn, B., Shu, C.-W., Johnson, C., Tadmor, E.: Essentially Non-oscillatory and Weighted Essentially Non-oscillatory Schemes for Hyperbolic Conservation Laws. Springer, Berlin (1998)

[5] Du J, Shu C-W, Zhong X. An improved simple WENO limiter for discontinuous Galerkin methods solving hyperbolic systems on unstructured meshes. J. Comput. Phys., 2022, 467 111424

[6] Dumbser M, Käser M. Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. J. Comput. Phys., 2007, 221(2): 693-723

[7] Feng Y, Liu T. A characteristic-featured shock wave indicator on unstructured grids based on training an artificial neuron. J. Comput. Phys., 2021, 443 110446

[8] Feng Y, Liu T, Wang K. A characteristic-featured shock wave indicator for conservation laws based on training an artificial neuron. J. Sci. Comput., 2020, 83: 1-34

[9] Friedrich O. Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids. J. Comput. Phys., 1998, 144(1): 194-212

[10] Harten A. Adaptive multiresolution schemes for shock computations. J. Comput. Phys., 1994, 115(2): 319-338

[11] Hill DJ, Pullin DI. Hybrid tuned center-difference-WENO method for large eddy simulations in the presence of strong shocks. J. Comput. Phys., 2004, 194(2): 435-450

[12] Hu C, Shu C-W. Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys., 1999, 150(1): 97-127

[13] Huang B, Qiu J. Hybrid WENO schemes with Lax-Wendroff type time discretization. J. Math. Study, 2017, 50(3): 242-267

[14] Jiang G-S, Shu C-W. Efficient implementation of weighted ENO schemes. J. Comput. Phys., 1996, 126(1): 202-228

[15] Kochi SSP, Ramakrishna M. A compact subcell WENO limiting strategy using immediate neighbours for Runge-Kutta discontinuous Galerkin methods. Int. J. Comput. Math., 2021, 98(3): 608-626

[16] Krivodonova L, Xin J, Remacle J-F, Chevaugeon N, Flaherty JE. Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws. Appl. Numer. Math., 2004, 48(3/4): 323-338

[17] Lax, P.D.: Weak solutions of nonlinear hyperbolic equations and their numerical computation. In: Selected Papers Volume I, pp. 198–232. Springer, Berlin (2005)

[18] Levy D, Puppo G, Russo G. Central WENO schemes for hyperbolic systems of conservation laws. ESAIM Math. Model. Numer. Anal., 1999, 33(3): 547-571

[19] Li G, Qiu J. Hybrid weighted essentially non-oscillatory schemes with different indicators. J. Comput. Phys., 2010, 229(21): 8105-8129

[20] Liu X-D, Osher S, Chan T. Weighted essentially non-oscillatory schemes. J. Comput. Phys., 1994, 115(1): 200-212

[21] Luo H, Baum JD, Löhner R. A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids. J. Comput. Phys., 2007, 225(1): 686-713

[22] Pirozzoli S. Conservative hybrid compact-WENO schemes for shock-turbulence interaction. J. Comput. Phys., 2002, 178(1): 81-117

[23] Qian C, Lu C, Liu L. A consistent and well-balanced hybrid weighted essentially non-oscillatory scheme for shallow water equations on unstructured meshes. Numer. Methods Partial Differ. Equ., 2025, 41(1): 23158

[24] Qiu J, Shu C-W. On the construction, comparison, and local characteristic decomposition for high-order central WENO schemes. J. Comput. Phys., 2002, 183(1187-209

[25] Qiu J, Shu C-W. Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: one-dimensional case. J. Comput. Phys., 2004, 193(1): 115-135

[26] Qiu J, Shu C-W. Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method II: two dimensional case. Comput. Fluids, 2005, 34(6): 642-663

[27] Ray D, Hesthaven JS. An artificial neural network as a troubled-cell indicator. J. Comput. Phys., 2018, 367: 166-191

[28] Ray D, Hesthaven JS. Detecting troubled-cells on two-dimensional unstructured grids using a neural network. J. Comput. Phys., 2019, 397 108845

[29] Shi J, Hu C, Shu C-W. A technique of treating negative weights in WENO schemes. J. Comput. Phys., 2002, 175(1): 108-127

[30] Shu, C.-W.: High order ENO and WENO schemes for computational fluid dynamics. In: High-order Methods for Computational Physics. Lecture Notes in Computational Science and Engineering, vol. 9, pp. 439–582. Springer, Berlin (1999)

[31] Shu C-W, Osher S. Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys., 1988, 77(2): 439-471

[32] Toro EF. Riemann Solvers and Numerical Methods for Fluid Dynamics: a Practical Introduction, 2013, Berlin, Springer

[33] Woodward P, Colella P. The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys., 1984, 54(1): 115-173

[34] Zhong X, Shu C-W. A simple weighted essentially nonoscillatory limiter for Runge-Kutta discontinuous Galerkin methods. J. Comput. Phys., 2013, 232(1): 397-415

[35] Zhu J, Qiu J. A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws. J. Comput. Phys., 2016, 318: 110-121

[36] Zhu J, Qiu J. A new type of modified WENO schemes for solving hyperbolic conservation laws. SIAM J. Sci. Comput., 2017, 39(3): 1089-1113

[37] Zhu J, Qiu J. New finite volume weighted essentially nonoscillatory schemes on triangular meshes. SIAM J. Sci. Comput., 2018, 40(2): 903-928

[38] Zhu J, Shu C-W, Qiu J. High-order Runge-Kutta discontinuous Galerkin methods with multi-resolution WENO limiters for solving steady-state problems. Appl. Numer. Math., 2021, 165: 482-499

[39] Zhu J, Zhong X, Shu C-W, Qiu J. Runge-Kutta discontinuous Galerkin method using a new type of WENO limiters on unstructured meshes. J. Comput. Phys., 2013, 248: 200-220

Funding

Innovative Research Group Project of the National Natural Science Foundation of China(42376018)

RIGHTS & PERMISSIONS

Shanghai University

Just Accepted

This article has successfully passed peer review and final editorial review, and will soon enter typesetting, proofreading and other publishing processes. The currently displayed version is the accepted final manuscript. The officially published version will be updated with format, DOI and citation information upon launch. We recommend that you pay attention to subsequent journal notifications and preferentially cite the officially published version. Thank you for your support and cooperation.