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Piecewise Polynomial Mapping Method and Corresponding WENO Scheme with Improved Resolution

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  23 November 2015

Qin Li ,
Pengxin Liu  and
Hanxin Zhang
Qin Li*
Affiliation:
State Key Laboratory of Aerodynamics, China Aerodynamic Research and Development Center, Mianyang, Sichuan, 621000, China National Laboratory of Computational Fluid Dynamics, Beijing University of Aeronautics and Astronautics, Beijing, 100191, China
Pengxin Liu*
Affiliation:
State Key Laboratory of Aerodynamics, China Aerodynamic Research and Development Center, Mianyang, Sichuan, 621000, China National Laboratory of Computational Fluid Dynamics, Beijing University of Aeronautics and Astronautics, Beijing, 100191, China
Hanxin Zhang
Affiliation:
State Key Laboratory of Aerodynamics, China Aerodynamic Research and Development Center, Mianyang, Sichuan, 621000, China National Laboratory of Computational Fluid Dynamics, Beijing University of Aeronautics and Astronautics, Beijing, 100191, China
*
*Corresponding author. Email addresses: [email protected] (Q. Li), [email protected] (P. Liu)
*Corresponding author. Email addresses: [email protected] (Q. Li), [email protected] (P. Liu)
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Abstract

Abstract. The method of mapping function was first proposed by Henrick et al. [J. Comput. Phys. 207:542-547 (2005)] to adjust nonlinear weights in [0,1] for the fifth order WENO scheme, and through which the requirement of convergence order is satisfied and the performance of the scheme is improved. Different from Henrick’s method, a concept of piecewise polynomial function is proposed in this study and corresponding WENO schemes are obtained. The advantage of the new method is that the function can have a gentle profile at the location of the linear weight (or the mapped nonlinear weight can be close to its linear counterpart), and therefore is favorable for the resolution enhancement. Besides, the function also has the flexibility of quick convergence to identity mapping near two endpoints of [0,1], which is favorable for improved numerical stability. The fourth-, fifth- and sixth-order polynomial functions are constructed correspondingly with different emphasis on aforementioned flatness and convergence. Among them, the fifth-order version has the flattest profile. To check the performance of the methods, the 1-D Shu-Osher problem, the 2-D Riemann problem and the double Mach reflection are tested with the comparison of WENO-M, WENO-Z and WENO-NS. The proposed new methods show the best resolution for describing shear-layer instability of the Riemann problem, and they also indicate high resolution in computations of double Mach reflection, where only these proposed schemes successfully resolved the vortex-pairing phenomenon. Other investigations have shown that the single polynomial mapping function has no advantage over the proposed piecewise one, and it is of no evident benefit to use the proposed method for the symmetric fifth-order WENO. Overall, the fifth-order piecewise polynomial and corresponding WENO scheme are suggested for resolution improvement.

Keywords

WENOmapping function

MSC classification

Secondary: 65M06: Finite difference methods 65M12: Stability and convergence of numerical methods
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 5 , November 2015 , pp. 1417 - 1444
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Liu, X.-D., Osher, S., Chan, T., Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115 (1994) 200–212.Google Scholar
[2]Jiang, G. S., Shu, C.-W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126 (1996) 202–228.Google Scholar
[3]Henrick, A. K., Aslam, T. D., J. M., , Powers, Mapped weigted essentially non-oscillatory schemes: achieving optimal order near critical points, J. Comput. Phys., 207 (2005) 542–567.CrossRefGoogle Scholar
[4]Ha, Y., Kim, C.-H., Lee, Y.-J., Yoon, J., An improved weighted essentially non-oscillatory scheme with a new smoothness indicator, J. Comput. Phys., 232 (2013) 68–86.Google Scholar
[5]Borges, R., Carmona, M., Costa, B., Don, W-S., An improved weighted essentially non-oscillatory scheme for hyperbolilc conservation laws, J. Comput. Phys., 227 (2008) 3191–3211.Google Scholar
[6]Lax, P. D. and Liu, X-D., Solution of two-dimensional Riemann problems of gas dynamics by positive schemes, SIAMJ. Sci. Comput., Vol. 19, No. 2 (1998) 319–340.Google Scholar
[7]Pirozzoli, S., Numerical methods for high-speed flows, Annu. Rev. Fluid Mech., 43 (2010) 163–194.Google Scholar
[8]Li, Q., Guo, Q.-L., Zhang, H.-X., Analyses on the dispersion overshoot and inverse dissipation of high order schemes, Adv. Appl. Math. Mech., 5 (2013) 809–824.Google Scholar
[9]Martn, M. P., Taylor, E. M., Wu, M., Weirs, V. G., A Bandwidth-Optimized WENO Scheme for the Direct Numerical Simulation of Compressible Turbulence, J. Comput. Phys., 220 (2006) 270–289.Google Scholar
[10]MacCormack, R. W., A numerical method for solving the equations of compressible viscous flow, AIAA J., 20 (1982).Google Scholar
[11]Gnoffo, P., CFD validation studies for hypersonic flow prediction, AIAA paper 2001–1025 (2001).Google Scholar
[12]Inoue, O. and Hattori, Y., Sound generation by shock-vortex interactions, J. Fluid Mech., 380 (1999) 81–116.Google Scholar