Abstract
An improved version of the three-level order-adaptive weighted essentially non-oscillatory (WENO-OA) scheme introduced in Neelan et al. (Results Appl Math 12:100217, 2021) is presented. The dependence of the WENO-OA scheme on the smoothness indicators of the Jiang–Shu WENO (WENO-JS) scheme is replaced with a new smoothness estimator with a smaller computational cost. In the present scheme, the smoothness indicator is only used to identify the smooth and non-smooth sub-stencils of the WENO scheme. The direct connection between the final WENO weights and smoothness indicators is decoupled so that it can exactly satisfy the Taylor expansion, which improves the accuracy of the scheme. The novel scheme denoted WENO-OA-I, is a three-level scheme because it can achieve the order of accuracy from three to five, while the classical scheme only achieves either the third or fifth order of accuracy. As a consequence of this property, the present scheme exhibits improved convergence rates. The performance of the new scheme is tested for hyperbolic equations with discontinuous solutions. The present scheme is up to 5.4 times computationally less expensive than the classical schemes.
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References
Aràndiga F, Baeza A, Belda AM, Mulet P (2011) Analysis of WENO schemes for full and global accuracy. SIAM J Numer Anal 49(2):893–915. https://doi.org/10.1137/100791579
Arora M, Roe PL (1997) A well-behaved TVD limiter for high-resolution calculations of unsteady flow. J Comput Phys 132(1):3–11. https://doi.org/10.1006/jcph.1996.5514
Bhise AA, Naga Raju G, Samala R, Devakar M (2019) An efficient hybrid WENO scheme with a problem independent discontinuity locator. Internat J Numer.Methods Fluids 91(1):1–28. https://doi.org/10.1002/fld.4739
Borges R, Carmona M, Costa B, Don WS (2008) An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J Comput Phys 227(6):3191–3211. https://doi.org/10.1016/j.jcp.2007.11.038
Guo Q, Sun D, Li C, Liu P, Zhang H (2020) A new discontinuity indicator for hybrid WENO schemes. J Sci Comput 83(2):1–33. https://doi.org/10.1007/s10915-020-01217-w
Hang T, Zhai Y, Zhou Z, Zhao W (2021) Conservative characteristic finite difference method based on ENO and WENO interpolation for 2D convection-diffusion equations. Comput Appl Math 40(6):202–21. https://doi.org/10.1007/s40314-021-01594-4
Henrick AK, Aslam TD, Powers JM (2005) Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points. J Comput Phys 207(2):542–567. https://doi.org/10.1016/j.jcp.2005.01.023
Hesthaven JS (2018) Numerical Methods for Conservation Laws. Comput Sci Eng 18: 570. https://doi.org/10.1137/1.9781611975109. (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA)
Jiang G-S, Shu C-W (1996) Efficient implementation of weighted ENO schemes. J Comput Phys 126(1):202–228. https://doi.org/10.1006/jcph.1996.0130
Kemm F (2016) On the proper setup of the double Mach reflection as a test case for the resolution of gas dynamics codes. Comput Fluids 132:72–75. https://doi.org/10.1016/j.compfluid.2016.04.008
Kumar R, Chandrashekar P (2018) Simple smoothness indicator and multi-level adaptive order WENO scheme for hyperbolic conservation laws. J Comput Phys 375:1059–1090. https://doi.org/10.1016/j.jcp.2018.09.027
Kundu A, De S (2017) Navier-Stokes simulation of shock-heavy bubble interaction: comparison of upwind and WENO schemes. Comput Fluids 157:131–145. https://doi.org/10.1016/j.compfluid.2017.08.025
Lax PD, Liu X-D (1998) Solution of two-dimensional Riemann problems of gas dynamics by positive schemes. SIAM J Sci Comput 19(2):319–340. https://doi.org/10.1137/S1064827595291819
LeVeque RJ (2002) Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics, p. 558. Cambridge University Press, Cambridge, UK. https://doi.org/10.1017/CBO9780511791253
Liu X-D, Osher S, Chan T (1994) Weighted essentially non-oscillatory schemes. J Comput Phys 115(1):200–212. https://doi.org/10.1006/jcph.1994.1187
Neelan AAG, Nair MT (2018) Hybrid finite difference-finite volume schemes on non-uniform grid. In: Singh MK, Kushvah BS, Seth GS, Prakash J (eds) Applications of Fluid Dynamics, pp. 329–340. Springer, Singapore. https://doi.org/10.1007/978-981-10-5329-0_24
Neelan AG, Nair M (2018) Hyperbolic Runge–Kutta method using evolutionary algorithm. J Comput Nonlin Dyn 13(11). https://doi.org/10.1115/1.4040708
Neelan AG, Nair MT (2020) Discontinuity preserving scheme. Int J Math Eng Manag Sci 5(4):631–642. https://doi.org/10.33889/IJMEMS.2020.5.4.051
Neelan AG, Nair MT (2020) Higher-order slope limiters for Euler equation. J Appl Comput Mech. https://doi.org/10.22055/jacm.2020.32845.2088
Neelan AAG, Nair MT, Bürger R (2021) Three-level order-adaptive weighted essentially non-oscillatory schemes. Results Appl Math 12:100217. https://doi.org/10.1016/j.rinam.2021.100217
Peer AAI, Dauhoo MZ, Bhuruth M (2009) A method for improving the performance of the WENO5 scheme near discontinuities. Appl Math Lett 22(11):1730–1733. https://doi.org/10.1016/j.aml.2009.05.016
Peng J, Zhai C, Ni G, Yong H, Shen Y (2019) An adaptive characteristic-wise reconstruction WENO-Z scheme for gas dynamic Euler equations. Comput Fluids 179:34–51. https://doi.org/10.1016/j.compfluid.2018.08.008
Qing F, Yu X, Jia Z, Li Z (2021) A cell-centered Lagrangian discontinuous Galerkin method using WENO and HWENO limiter for compressible Euler equations in two dimensions. Comput Appl Math 40(6):212–33. https://doi.org/10.1007/s40314-021-01575-7
Rajpoot MK, Sengupta TK, Dutt PK (2010) Optimal time advancing dispersion relation preserving schemes. J Comput Phys 229(10):3623–3651. https://doi.org/10.1016/j.jcp.2010.01.018
Rathan S, Raju GN (2017) An improved non-linear weights for seventh-order weighted essentially non-oscillatory scheme. Comput Fluids 156:496–514. https://doi.org/10.1016/j.compfluid.2017.08.023
Rathan S, Raju GN (2018) Improved weighted ENO scheme based on parameters involved in nonlinear weights. Appl Math Comput 331:120–129. https://doi.org/10.1016/j.amc.2018.03.034
Shen Y, Zha G (2014) Improvement of weighted essentially non-oscillatory schemes near discontinuities. Comput Fluids 96:1–9. https://doi.org/10.1016/j.compfluid.2014.02.010
Shu C-W (2009) High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM Rev 51(1):82–126. https://doi.org/10.1137/070679065
Shu C-W (2020) Essentially non-oscillatory and weighted essentially non-oscillatory schemes. Acta Numer. 29:701–762. https://doi.org/10.1017/S0962492920000057
Shu C-W, Osher S (1989) Efficient implementation of essentially nonoscillatory shock-capturing schemes. II. J Comput Phys 83(1):32–78. https://doi.org/10.1016/0021-9991(89)90222-2
Sod GA (1978) A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J Comput Phys 27(1):1–31. https://doi.org/10.1016/0021-9991(78)90023-2
Toro EF (2009) Riemann solvers and numerical methods for fluid dynamics: a practical introduction, 3rd edn. Springer, Berlin
van Lith BS, ten Thije Boonkkamp JHM, Ijzerman WL (2017) Embedded WENO: a design strategy to improve existing WENO schemes. J Comput Phys 330:529–549. https://doi.org/10.1016/j.jcp.2016.11.026
Woodward P, Colella P (1984) The numerical simulation of two-dimensional fluid flow with strong shocks. J Comput Phys 54(1):115–173. https://doi.org/10.1016/0021-9991(84)90142-6
Zhang Y-T, Shu C-W (2016) ENO and WENO schemes. In: Handbook of Numerical Methods for Hyperbolic Problems. Handb. Numer. Anal., vol. 17, pp. 103–122. Elsevier/North-Holland, Amsterdam
Zhao Z, Zhu J, Chen Y, Qiu J (2019) A new hybrid WENO scheme for hyperbolic conservation laws. Comput Fluids 179:422–436. https://doi.org/10.1016/j.compfluid.2018.10.024
Acknowledgements
R.B. acknowledges support by project MATH-Amsud 22-MATH-05 “NOTION: NOn-local conservaTION laws for engineering, biological and epidemiological applications: theoretical and numerical” and from ANID (Chile) through Fondecyt project 1210610; Anillo project ANID/ACT210030; Centro de Modelamiento Matemático (CMM), project FB210005 of BASAL funds for Centers of Excellence; and CRHIAM, project ANID/FONDAP/15130015.
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Neelan, A.A.G., Chandran, R.J., Diaz, M.A. et al. An efficient three-level weighted essentially non-oscillatory scheme for hyperbolic equations. Comp. Appl. Math. 42, 70 (2023). https://doi.org/10.1007/s40314-023-02214-z
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DOI: https://doi.org/10.1007/s40314-023-02214-z