Hybrid Finite Difference Weighted Essentially Non-oscillatory Schemes for the Compressible Ideal Magnetohydrodynamics Equation
- 167 Downloads
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.
KeywordsWENO 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.
- 11.Fitzpatrick, R.: Plasma Physics: An Introduction. CRC Press, Boca Raton (2015)Google Scholar
- 31.Rossmanith, J.: A wave propagation method with constrained transport for shallow water and ideal magnetohydrodynamics, Ph.D Thesis, University of Washington, Seattle (2002)Google Scholar
- 34.Shu, C.W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In: Quarteroni, A. (ed.) Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, vol. 1697, pp. 325–432. Springer, Berlin (1998)CrossRefGoogle Scholar