Abstract
PHM methods have been used successfully as reconstruction procedures to design high-order Riemann solvers for nonlinear scalar and systems of conservation laws, (see [8], [1], [4]). We introduce a new class of polynomial reconstruction procedures based on the harmonic mean of the absolute values of finite diferences used as difference-limiter, following the original idea used before to design the piecewise hyperbolic method, introduced in [8]. We call those methods ’harmonic ENO methods’, (HENO). Furthermore, we give analytical and numerical evidence of the good behavior of these methods used as reconstruction procedures for the numerical approximation by means of shock-capturing methods for scalar and systems of conservation laws in ID. We discuss, in particular, the behavior of a fourth order harmonic ENO method,(HEN04 in short), compared with PHM, EN03 and WEN05 methods, (see [2], [10], [3]).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. Donat and A. Marquina, Capturing Shock Reflections: An improved Flux Formula J. Comput. Phys., v. 125, (1996) pp. 42–58.
A. Harten, B. Engquist, S. Osher and S. Chahravarthy, Uniformly High Order Accurate Essentially Non-oscillatory Schemes III, J. Comput. Phys., v. 71, No. 2, (1987), pp. 231–303.
O.S. Jiang and C. W. Shu, Efficient Implementation of weighted ENO schemes, J. Comput. Phys., 126, (1996), p. 202.
S. Li and L. Petzold, Moving Mesh Methods with Upwinding Schemes for Time-Dependent PDEs, J. Comput. Phys., v. 131, (1997), pp. 368–377.
R.J. LeVeque Numerical methods for Conservation Laws, Birkhauser Verlag, Zuerich, (1990).
X-D. Liu and S. Osher and T. Chan Weighted essentially non-oscillatory schemes, J. Comput. Phys., v. 115, (1994), pp. 200–212.
A. Rogertson and E. Meiburg, A Numerical Study of the Convergence of ENO schemes, J. Sci. Comp., v. 5, (1990) pp. 127–150.
A. Marquina, Local Piecewise Hyperbolic Reconstructions for Nonlinear Scalar Conservation Laws, SIAM J. Sci. Comp., v. 15, (1994) pp. 892–915.
S. Serna and A. Marquina, Power ENO methods, preprint.
C. W. Shu and S. J. Osher, Efficient Implementation of Essentially Non-Oscillatory Shock Capturing Schemes II, J. Comput. Phys., v. 83, (1989) pp. 32–78.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Marquina, A., Serna, S. (2003). Afternotes on PHM: Harmonic ENO Methods. In: Hou, T.Y., Tadmor, E. (eds) Hyperbolic Problems: Theory, Numerics, Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55711-8_67
Download citation
DOI: https://doi.org/10.1007/978-3-642-55711-8_67
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-62929-7
Online ISBN: 978-3-642-55711-8
eBook Packages: Springer Book Archive