Abstract
Fu et al. (J Comput Phys 374:724–751, 2018), a class of adaptive targeted ENO (TENO) schemes have been proposed. Excellent resolution and robustness of the TENO scheme were validated by benchmark problems, but the computational cost is high. In order to improve the computational efficiency, a modified TENO scheme is proposed in this article. First, using the weighting strategy of the fifth-order TENO as an explicit discontinuity detector, locations of discontinuities are detected on the five-point stencil, and the initial target stencil with maximum support on the five-point stencil is obtained. Then, with a simple discontinuity detection strategy, the initial target stencil is enlarged point by point in the direction where no discontinuity is detected. In this manner, final target stencil with maximum support, not crossed by discontinuities, on the full six- or eight-point stencil is formed. Unlike TENO using candidate stencils with incrementally increased width, all stencils are three-point in the new framework. Due to the simplified determination of final target stencil and avoiding expensive calculation of smoothness indicator of large stencil, the computational cost of TENO-M is significantly lower than that of TENO. Several benchmark problems are conducted to validate the performance of the proposed scheme.
Similar content being viewed by others
References
Pirozzoli, S.: Numerical methods for high-speed flows. Annu. Rev. Fluid Mech. 43, 163–194 (2011). https://doi.org/10.1146/annurev-fluid-122109-160718
Jameson, A.: Analysis and design of numerical schemes for gas dynamics 1: artificial diffusion, upwind biasing, limiters and their effect on accuracy and multigrid convergence. Int. J. Comput. Fluid Dyn. 4, 171–218 (1995). https://doi.org/10.1080/10618569508904524
Harten, A.: A high resolution scheme for the computation of weak solutions of hyperbolic conservation laws. J. Comput. Phys. 49, 357–393 (1983)
Harten, A., Engquist, B., Osher, S., Chakravarthy, S.R.: Uniformly high order accurate essentially non-oscillatory schemes, III. J. Comput. Phys. 71, 231–303 (1987). https://doi.org/10.1016/0021-9991(87)90031-3
Liu, X.D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200–212 (1994)
Fu, L., Hu, X.Y., Adams, N.A.: A family of high-order targeted ENO schemes for compressible-fluid simulations. J. Comput. Phys. 305, 333–359 (2016). https://doi.org/10.1016/j.jcp.2015.10.037
Jiang, G.S., Shu, C.W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996). https://doi.org/10.1006/jcph.1996.0130
Henrick, A.K., Aslam, T.D., Powers, J.M.: Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points. J. Comput. Phys. 207, 542–567 (2005). https://doi.org/10.1016/j.jcp.2005.01.023
Borges, R., Carmona, M., Costa, B., Don, W.S.: An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J. Comput. Phys. 227, 3191–3211 (2008). https://doi.org/10.1016/j.jcp.2007.11.038
Don, W.S., Borges, R.: Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes. J. Comput. Phys. 250, 347–372 (2013). https://doi.org/10.1016/j.jcp.2013.05.018
Zhao, S., Lardjane, N., Fedioun, I.: Comparison of improved finite-difference WENO schemes for the implicit large eddy simulation of turbulent non-reacting and reacting high-speed shear flows. Comput. Fluids 95, 74–87 (2014). https://doi.org/10.1016/j.compfluid.2014.02.017
Martín, M.P., Taylor, E.M., Wu, M., Weirs, V.G.: A bandwidth-optimized WENO scheme for the effective direct numerical simulation of compressible turbulence. J. Comput. Phys. 220, 270–289 (2006). https://doi.org/10.1016/j.jcp.2006.05.009
Hu, X.Y., Wang, Q., Adams, N.A.: An adaptive central-upwind weighted essentially non-oscillatory scheme. J. Comput. Phys. 229, 8952–8965 (2010). https://doi.org/10.1016/j.jcp.2010.08.019
Hu, X.Y., V.K. Tritschler, Pirozzoli, S., Adams, N.A.: Dispersion-dissipation condition for finite difference schemes
Suresh, A., Huynh, H.T.: Accurate monotonicity-preserving schemes with Runge–Kutta time stepping. J. Comput. Phys. 136, 83–99 (1997). https://doi.org/10.1006/jcph.1997.5745
Gerolymos, G.A., Sénéchal, D., Vallet, I.: Very-high-order weno schemes. J. Comput. Phys. 228, 8481–8524 (2009). https://doi.org/10.1016/j.jcp.2009.07.039
Zhang, X., Shu, C.W.: Positivity-preserving high order finite difference WENO schemes for compressible Euler equations. J. Comput. Phys. 231, 2245–2258 (2012). https://doi.org/10.1016/j.jcp.2011.11.020
Fu, L., Hu, X.Y., Adams, N.A.: Targeted ENO schemes with tailored resolution property for hyperbolic conservation laws. J. Comput. Phys. 349, 97–121 (2017). https://doi.org/10.1016/j.jcp.2017.07.054
Fu, L.: A very-high-order TENO scheme for all-speed gas dynamics and turbulence. Comput. Phys. Commun. 244, 117–131 (2019). https://doi.org/10.1016/j.cpc.2019.06.013
Fu, L., Hu, X.Y., Adams, N.A.: A new class of adaptive high-order targeted ENO schemes for hyperbolic conservation laws. J. Comput. Phys. 374, 724–751 (2018). https://doi.org/10.1016/j.jcp.2018.07.043
Fu, L.: A hybrid method with teno based discontinuity indicator for hyperbolic conservation laws. Commun. Comput. Phys. 26, 973–1007 (2019). https://doi.org/10.4208/cicp.OA-2018-0176
Pirozzoli, S.: On the spectral properties of shock-capturing schemes. J. Comput. Phys. 219, 489–497 (2006). https://doi.org/10.1016/j.jcp.2006.07.009
Balsara, D.S., Shu, C.W.: Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys. (2000). https://doi.org/10.1006/jcph.2000.6443
Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43, 357–372 (1981). https://doi.org/10.1006/jcph.1997.5705
Yamaleev, N.K., Carpenter, M.H.: A systematic methodology for constructing high-order energy stable WENO schemes. J. Comput. Phys. 228, 4248–4272 (2009). https://doi.org/10.1016/j.jcp.2009.03.002
Sod, G.A.: A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys. 27, 1–31 (1978). https://doi.org/10.1016/0021-9991(78)90023-2
Lax, P.D.: Weak solutions of nonlinear hyperbolic equations and their numerical computation. Commun. Pure Appl. Math. 7, 159–193 (1954). https://doi.org/10.1002/cpa.3160070112
Woodward, P., Colella, P.: The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54, 115–173 (1984). https://doi.org/10.1016/0021-9991(84)90142-6
Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory schemes II. J. Comput. Phys. 83, 32–78 (1989)
Xu, Z., Shu, C.W.: Anti-diffusive flux corrections for high order finite difference WENO schemes. J. Comput. Phys. 205, 458–485 (2005). https://doi.org/10.1016/j.jcp.2004.11.014
van Leer, B.: Upwind and High-Resolution Schemes. Springer, Berlin (1997)
Lax, P.D., Liu, X.D.: Solution of two-dimensional Riemann problems of gas dynamics by positive schemes. SIAM J. Sci. Comput. 19, 319–340 (1998). https://doi.org/10.1137/S1064827595291819
Hong, Z., Ye, Z., Ye, K.: An optimised five-point-stencil weighted compact nonlinear scheme for hyperbolic conservation laws. Int. J. Comput. Fluid Dyn. 35, 179–196 (2021). https://doi.org/10.1080/10618562.2021.1906419
Samtaney, R., Pullin, D.I., Kosović, B.: Direct numerical simulation of decaying compressible turbulence and shocklet statistics. Phys. Fluids 13, 1415–1430 (2001). https://doi.org/10.1063/1.1355682
Acknowledgements
The authors would like to acknowledge the funding support of this research by National Natural Science Foundation of China (No. 11732013).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Hong, Z., Ye, Z. & Ye, K. A Modified TENO Scheme with Improved Efficiency. J Sci Comput 91, 37 (2022). https://doi.org/10.1007/s10915-022-01809-8
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10915-022-01809-8