A New Sixth-Order WENO Scheme for Solving Hyperbolic Conservation Laws

doi:10.1007/s42967-020-00112-3

Communications on Applied Mathematics and Computation ›› 2023, Vol. 5 ›› Issue (1): 3-30.doi: 10.1007/s42967-020-00112-3

• ORIGINAL PAPERS • Previous Articles Next Articles

A New Sixth-Order WENO Scheme for Solving Hyperbolic Conservation Laws

Kunlei Zhao^1,2, Yulong Du³, Li Yuan^1,2

1 State Key Laboratory of Scientific and Engineering Computing(LSEC) and Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;
2 School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China;
3 School of Mathematical Sciences, Beihang University, Beijing 100191, China

Received:2020-09-04 Revised:2020-11-19 Online:2023-03-20 Published:2023-03-08
Contact: Li Yuan,E-mail:lyuan@lsec.cc.ac.cn;Kunlei Zhao,E-mail:klzhao@lsec.cc.ac.cn;Yulong Du,E-mail:kunyu0918@163.com E-mail:lyuan@lsec.cc.ac.cn;klzhao@lsec.cc.ac.cn;kunyu0918@163.com
Supported by:
This work is supported by the National Natural Science Foundation of China (91641107, 91852116, 12071470) and Fundamental Research of Civil Aircraft (MJ-F-2012-04) of Ministry of Industrialization and Information of China.

Abstract

Abstract: In this paper, we develop a new sixth-order WENO scheme by adopting a convex combination of a sixth-order global reconstruction and four low-order local reconstructions. Unlike the classical WENO schemes, the associated linear weights of the new scheme can be any positive numbers with the only requirement that their sum equals one. Further, a very simple smoothness indicator for the global stencil is proposed. The new scheme can achieve sixthorder accuracy in smooth regions. Numerical tests in some one- and two-dimensional benchmark problems show that the new scheme has a little bit higher resolution compared with the recently developed sixth-order WENO-Z6 scheme, and it is more efficient than the classical fifth-order WENO-JS5 scheme and the recently developed sixth-order WENO6-S scheme.

Key words: Global smoothness indicator, Linear weights, Sixth-order accuracy, WENO

CLC Number:

Kunlei Zhao, Yulong Du, Li Yuan. A New Sixth-Order WENO Scheme for Solving Hyperbolic Conservation Laws[J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 3-30.

TrendMD

References

1. Acker, F., Borges, R., Costa, B.: An improved WENO-Z scheme. J. Comput. Phys. 313, 726–753 (2016)
2. Balsara, D.S., Garain, S.K., Shu, C.-W.: An efficient class of WENO schemes with adaptive order. J. Comput. Phys. 326, 780–804 (2016)
3. 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)
4. Castro, M., Costa, B., Don, W.S.: High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws. J. Comput. Phys. 230, 1766–1792 (2011)
5. Donat, R., Marquina, A.: Capturing shock reflections: an improved flux formula. J. Comput. Phys. 125, 42–58 (1996)
6. Fardipour, K., Mansour, K.: A modified seventh-order WENO scheme with new nonlinear weights for hyperbolic conservation laws. Comput. Math. Appl. 78, 3748–3769 (2019)
7. Feng, H., Hu, F.X., Wang, R.: A new mapped weighted essentially non-oscillatory scheme. J. Sci. Comput. 51, 449–473 (2012)
8. Feng, H., Huang, C., Wang, R.: An improved mapped weighted essentially non-oscillatory scheme. Appl. Math. Comput. 232, 453–468 (2014)
9. 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 (2007)
10. Hu, F.X.: The 6th-order weighted ENO schemes for hyperbolic conservation laws. Comput. Fluids 174, 34–45 (2018)
11. Hu, X.Y., Wang, Q., Adams, N.A.: An adaptive central-upwind weighted essentially non-oscillatory scheme. J. Comput. Phys. 229, 8952–8965 (2010)
12. Huang, C., Chen, L.L.: A new adaptively central-upwind sixth-order WENO scheme. J. Comput. Phys. 357, 1–15 (2018)
13. Jiang, G.S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)
14. Jiang, G.S., Wu, C.C.: A high-order WENO finite difference scheme for the equations of ideal magnetohydrodynamics. J. Comput. Phys. 150, 561–594 (1999)
15. Lax, P.D.: Weak solutions of nonlinear hyperbolic equations and their numerical computation. Commun. Pure Appl. Math. 7, 159–193 (1954)
16. Levy, D., Puppo, G., Russo, G.: Central WENO schemes for hyperbolic systems of conservation laws. Math. Model. Numer. Anal. 33, 547–571 (1999)
17. Liu, X.D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200–212 (1994)
18. Martin, 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)
19. Nonomura, T., Kitamura, K., Fujii, K.: A simple interface sharpening technique with a hyperbolic tangent function applied to compressible two-fluid modeling. J. Comput. Phys. 258, 95–117 (2014)
20. Peng, J., Zhai, C.L., Ni, G.X., Yong, H., Shen, Y.Q.: An adaptive characteristic-wise reconstruction WENO-Z scheme for gas dynamic Euler equations. Comput. Fluids 179, 34–51 (2019)
21. Schulz-Rinne, C.-W., Collins, J.P., Glaz, H.M.: Numerical solution of the Riemann problem for twodimensional gas dynamics. SIAM J. Sci. Comput. 14, 1394–1414 (1993)
22. Shu, C.-W.: Total-variation-diminishing time discretizations. SIAM J. Sci. Stat. Comput. 9, 1073–1084 (1988)
23. Shu, C.-W., Osher, S.: Efficient implement of essentially non-oscillatory shock capturing schemes. II. J. Comput. Phys. 83, 32–78 (1989)
24. Sod, G.A.: A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys. 27, 1–31 (1978)
25. Sun, Z.Y., Inaba, S., Xiao, F.: Boundary variation diminishing (BVD) reconstruction: a new approach to improve Godunov schemes. J. Comput. Phys. 322, 309–325 (2016)
26. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics, pp. 87–114. Springer, Berlin (2013)
27. Wang, Y.H., Du, Y.L., Zhao, K.L., Yuan, L.: A new 6th-order WENO scheme with modified stencils. Comput. Fluids 208, 104625 (2020)
28. Woodward, P.R., Colella, P.: The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54, 115–173 (1984)
29. Zhu, J., Qiu, J.X.: A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws. J. Comput. Phys. 318, 110–121 (2016)

[1]	Jun Zhu, Jianxian Qiu. New Finite Difference Mapped WENO Schemes with Increasingly High Order of Accuracy [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 64-96.
[2]	Scott E. Field, Sigal Gottlieb, Zachary J. Grant, Leah F. Isherwood, Gaurav Khanna. A GPU-Accelerated Mixed-Precision WENO Method for Extremal Black Hole and Gravitational Wave Physics Computations [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 97-115.
[3]	Andrew Christlieb, Matthew Link, Hyoseon Yang, Ruimeng Chang. High-Order Semi-Lagrangian WENO Schemes Based on Non-polynomial Space for the Vlasov Equation [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 116-142.
[4]	M. Semplice, E. Travaglia, G. Puppo. One- and Multi-dimensional CWENOZ Reconstructions for Implementing Boundary Conditions Without Ghost Cells [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 143-169.
[5]	Linjin Li, Jingmei Qiu, Giovanni Russo. A High-Order Semi-Lagrangian Finite Difference Method for Nonlinear Vlasov and BGK Models [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 170-198.
[6]	Yifei Wan, Yinhua Xia. A New Hybrid WENO Scheme with the High-Frequency Region for Hyperbolic Conservation Laws [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 199-234.
[7]	Bao-Shan Wang, Wai Sun Don, Alexander Kurganov, Yongle Liu. Fifth-Order A-WENO Schemes Based on the Adaptive Diffusion Central-Upwind Rankine-Hugoniot Fluxes [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 295-314.
[8]	Xucheng Meng, Yaguang Gu, Guanghui Hu. A Fourth-Order Unstructured NURBS-Enhanced Finite Volume WENO Scheme for Steady Euler Equations in Curved Geometries [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 315-342.
[9]	G. Puppo, M. Semplice, G. Visconti. Quinpi: Integrating Conservation Laws with CWENO Implicit Methods [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 343-369.
[10]	Liang Li, Jun Zhu, Chi-Wang Shu, Yong-Tao Zhang. A Fixed-Point Fast Sweeping WENO Method with Inverse Lax-Wendroff Boundary Treatment for Steady State of Hyperbolic Conservation Laws [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 403-427.
[11]	Zepeng Liu, Yan Jiang, Mengping Zhang, Qingyuan Liu. High Order Finite Difference WENO Methods for Shallow Water Equations on Curvilinear Meshes [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 485-528.
[12]	Rathan Samala, Biswarup Biswas. Arc Length-Based WENO Scheme for Hamilton-Jacobi Equations [J]. Communications on Applied Mathematics and Computation, 2021, 3(3): 481-496.
[13]	Jun Zhu, Chi, Wang Shu. Convergence to Steady-State Solutions of the New Type of High-Order Multi-resolution WENO Schemes: a Numerical Study [J]. Communications on Applied Mathematics and Computation, 2020, 2(3): 429-460.

A New Sixth-Order WENO Scheme for Solving Hyperbolic Conservation Laws

Knowledge

Abstract

Cite this article

share this article

References

Related Articles 13

Recommended Articles

Metrics

Comments