Journal of Scientific Computing

, Volume 74, Issue 2, pp 607–630 | Cite as

Hybrid Finite Difference Weighted Essentially Non-oscillatory Schemes for the Compressible Ideal Magnetohydrodynamics Equation

Review Paper


In this paper, we present hybrid weighted essentially non-oscillatory (WENO) schemes with several discontinuity detectors for solving the compressible ideal magnetohydrodynamics (MHD) equation. Li and Qiu (J Comput Phys 229:8105–8129, 2010) examined effectiveness and efficiency of several different troubled-cell indicators in hybrid WENO methods for Euler gasdynamics. Later, Li et al. (J Sci Comput 51:527–559, 2012) extended the hybrid methods for solving the shallow water equations with four better indicators. Hybrid WENO schemes reduce the computational costs, maintain non-oscillatory properties and keep sharp transitions for problems. The numerical results of hybrid WENO-JS/WENO-M schemes are presented to compare the ability of several troubled-cell indicators with a variety of test problems. The focus of this paper, we propose optimal and reliable indicators for performance comparison of hybrid method using troubled-cell indicators for efficient numerical method of ideal MHD equations. We propose a modified ATV indicator that uses a second derivative. It is advantageous for differential discontinuity detection such as jump discontinuity and kink. A detailed numerical study of one-dimensional and two-dimensional cases is conducted to address efficiency (CPU time reduction and more accurate numerical solution) and non-oscillatory property problems. We demonstrate that the hybrid WENO-M scheme preserves the advantages of WENO-M and the ratio of computational costs of hybrid WENO-M and hybrid WENO-JS is smaller than that of WENO-M and WENO-JS.


WENO approximation Mapped WENO Hybrid WENO Ideal MHD Troubled-cell indicators 



Youngsoo Ha and Chang Ho Kim were supported by National R&D Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2014M1A7A1A03029872), also Youngsoo Ha was supported by the National Research Foundation of Korea (NRF) (NRF-2013R1A1A2013793). Kwang-Il You is supported by Ministry of Science, ICT and Future Planning of Korea (NIST Code: 1711046884). We would like to thank the anonymous reviewers for constructive suggestions to improve this paper.


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Computer Engineering, Glocal CampusKonkuk UniversityChungjuRepublic of Korea
  2. 2.National Fusion Research InstituteDaejeonRepublic of Korea
  3. 3.Department of Mathematical SciencesSeoul National UniversitySeoulRepublic of Korea

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