Abstract
This paper is devoted to the construction and analysis of a new smoothness index for WENO interpolation capable of dealing with corner discontinuities. The new smoothness index presented is initially developed for the point-value framework of Harten’s multiresolution. Even so, the ideas about how to extend the results to the cell-average framework are presented. The new smoothness index is inspired by the one proposed in Jiang and Shu (J Comput Phys 126(1):202–228, 1996). This index works very well for jump discontinuities as it was originally designed for the context of conservation laws in order to deal with problems that contain shocks and complicated fluid-structure interactions. Even so, it is easy to check that the mentioned index does not provide an appropriate performance for corner discontinuities. Our aim is to rise the order of accuracy of WENO interpolation near corner discontinuities. In order to do so, we will modify the original smoothness index proposed by Jiang and Shu such that the discontinuities in the first derivative of the function contribute effectively to the index. The modification proposed will produce a variation in the weights of WENO when dealing with a corner, that do not appear when using the smoothness indexes proposed by Jiang and Shu. The variation in the weights induced by the modification of the smoothness index will allow adaption to corner discontinuities, maintaining the adaption to jumps provided by the original smoothness index proposed by Jiang and Shu. The strategy proposed in Aràndiga et al. (SIAM J Numer Anal 49(2):893–915, 2011) can be adapted such that the accuracy is maintained near critical points at smooth zones.
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Acknowledgements
We would like to thank Professor Vicente Candela from University of Valencia for his useful comments about WENO and the new smoothness index proposed and also for his encouragement to finish this work. We would also like to thank the referees for their useful suggestions and comments that, with no doubt, have helped to improve the quality of this paper.
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S. Amat: The author have been supported through the Programa de Apoyo a la investigación de la fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 19374/PI714 and through the national research project MTM2015-64382-P (MINECO/FEDER).
J. Ruiz: The author have been supported through the Programa de Apoyo a la investigación de la fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 19374/PI714, through the national research project MTM2015-64382-P (MINECO/FEDER) and by the Fundación Seneca through the young researchers program Jiménez de la Espada.
Appendix
Appendix
In this Section we show how to obtain the smoothness indicator proposed in (20) in the point-values. We will also use Taylor expansions in order to show that these smoothness indexes can be written as \(D\left( 1+O(h^2)\right) \), where D is a constant that depends on h.
The polynomials that we are going to use are those in (28), (29), (30). Applying expression (20) to the stencils \(S_j=\{x_{j-3}, x_{j-2}, x_{j-1}, x_{j}\}\), \(S_{j+1}=\{x_{j-2}, x_{j-1}, x_{j}, x_{j+1}\}\) and \(S_{j+2}=\{x_{j-1}, x_{j}, x_{j+1}, x_{j+2}\}\) that correspond to the point-values \(\{f_{j-3}, f_{j-2}, f_{j-1}, f_j, f_{j+1}, f_{j+2}\}\), we obtain the smootness indexes in (21), (22), (23), that expanded take the expression,
Now we can substitute the Taylor expansions (34) in (64), (65), (66), and we obtain,
that is the desired result.
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Amat, S., Ruiz, J. New WENO Smoothness Indicators Computationally Efficient in the Presence of Corner Discontinuities. J Sci Comput 71, 1265–1302 (2017). https://doi.org/10.1007/s10915-016-0342-2
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DOI: https://doi.org/10.1007/s10915-016-0342-2
Keywords
- WENO
- New smoothness indicators
- Corner discontinuities
- Multiresolution schemes
- Improved adaption to discontinuities
- Signal processing