# Smoothness indicators for WENO scheme using undivided differences

## Main Article Content

## Abstract

The weighted essentially non-oscillatory method (WENO) has been used widely in numerical solutions during the last two decades. This method relies on Smoothness Indicators (SI) to produce smooth solutions near discontinuities. It was concluded before that evaluating SI based on undivided differences (UD) is inefficient, and the $L2$ norm of the interpolation polynomials was used instead. In the current study the idea of using UD is revisited with the key feature of careful selection of the stencil. Improvement in terms of the accuracy and the number of arithmetic operations is illustrated by numerical simulations.

## Article Details

How to Cite

KASEM, Tamer H. M. A.; SCHMITT, Francois.
Smoothness indicators for WENO scheme using undivided differences.

**Proceedings of the Conference Algoritmy**, [S.l.], p. 155-164, feb. 2016. Available at: <http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/404>. Date accessed: 22 sep. 2017.
Section

Articles

## References

[1] G. B. Whitham Linear and nonlinear waves John Wiley and sons, 1974.

[2] C. W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, Lect. Notes Math., 1697 (1998), pp. 325–432.

[3] X.-D. Liu, S. Osher, and T. Chan, Weighted Essentially Non-oscillatory Schemes, J. Comput. Phys., 115 (1994), pp. 200–212.

[4] G.-S. Jiang, C.-W. Shu, Efficient Implementation of Weighted ENO Schemes, J. Comput. Phys., 126 (1996), pp. 202–228.

[5] R. Borges, M. Carmona, B. Costa, and W.S. Don, An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws, J. Comput. Phys., 227 (2008), pp. 3191-3211.

[6] A.K. Henrick, T.D. Aslam, and J.M. Powers, Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points, J. Comput. Phys., 207 (2005), pp. 542-567.

[7] R. Archibald, A. Gelb, and J. Yoon, Polynomial Fitting for Edge Detection in Irregularly Sampled Signals and Images, SIAM J. Numer. Anal., 43 (2005), pp. 259–279.

[8] John A. Trangenstein, Numerical Solution of Hyperbolic Partial Differential Equations, Cambridge University Press, 2009.

[9] D.D. KNIGHT, Elements of Numerical Methods for Compressible Flows, Cambridge University Press, 2006.

[10] E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer, 2009.

[2] C. W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, Lect. Notes Math., 1697 (1998), pp. 325–432.

[3] X.-D. Liu, S. Osher, and T. Chan, Weighted Essentially Non-oscillatory Schemes, J. Comput. Phys., 115 (1994), pp. 200–212.

[4] G.-S. Jiang, C.-W. Shu, Efficient Implementation of Weighted ENO Schemes, J. Comput. Phys., 126 (1996), pp. 202–228.

[5] R. Borges, M. Carmona, B. Costa, and W.S. Don, An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws, J. Comput. Phys., 227 (2008), pp. 3191-3211.

[6] A.K. Henrick, T.D. Aslam, and J.M. Powers, Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points, J. Comput. Phys., 207 (2005), pp. 542-567.

[7] R. Archibald, A. Gelb, and J. Yoon, Polynomial Fitting for Edge Detection in Irregularly Sampled Signals and Images, SIAM J. Numer. Anal., 43 (2005), pp. 259–279.

[8] John A. Trangenstein, Numerical Solution of Hyperbolic Partial Differential Equations, Cambridge University Press, 2009.

[9] D.D. KNIGHT, Elements of Numerical Methods for Compressible Flows, Cambridge University Press, 2006.

[10] E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer, 2009.