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Piecewise Polynomial Mapping Method and Corresponding WENO Scheme withImproved 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. Emailaddresses: [email protected] (Q.Li), [email protected](P. Liu)
*Corresponding author. Emailaddresses: [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 thefifth order WENO scheme, and through which the requirement of convergence orderis satisfied and the performance of the scheme is improved. Different fromHenrick’s method, a concept of piecewise polynomial function isproposed in this study and corresponding WENO schemes are obtained. Theadvantage of the new method is that the function can have a gentle profile atthe location of the linear weight (or the mapped nonlinear weight can be closeto its linear counterpart), and therefore is favorable for the resolutionenhancement. Besides, the function also has the flexibility of quick convergenceto identity mapping near two endpoints of [0,1], which is favorable for improvednumerical stability. The fourth-, fifth- and sixth-order polynomial functionsare constructed correspondingly with different emphasis on aforementionedflatness and convergence. Among them, the fifth-order version has the flattestprofile. To check the performance of the methods, the 1-D Shu-Osher problem, the2-D Riemann problem and the double Mach reflection are tested with thecomparison of WENO-M, WENO-Z and WENO-NS. The proposed new methods show the bestresolution for describing shear-layer instability of the Riemann problem, andthey also indicate high resolution in computations of double Mach reflection,where only these proposed schemes successfully resolved the vortex-pairingphenomenon. Other investigations have shown that the single polynomial mappingfunction has no advantage over the proposed piecewise one, and it is of noevident benefit to use the proposed method for the symmetric fifth-order WENO.Overall, the fifth-order piecewise polynomial and corresponding WENO scheme aresuggested 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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