Skip to main content
SpringerLink
Log in

New WENO Smoothness Indicators Computationally Efficient in the Presence of Corner Discontinuities

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Jiang, G., Shu, C.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126(1), 202–228 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aràndiga, F., Baeza, A., Belda, A.M., Mulet, P.: Analysis of WENO schemes for full and global accuracy. SIAM J. Numer. Anal. 49(2), 893–915 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Harten, A.: Multiresolution representation of data II. SIAM J. Numer. Anal. 33(3), 1205–1256 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Harten, A.: Multiresolution representation of data: a general framework. SIAM J. Numer. Anal. 33(3), 1205–1256 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Harten, A., Osher, S.: Uniformly high-order accurate nonoscillatory schemes. I. SIAM J. Numer. Anal. 24(2), 279–309 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  6. Harten, A., Engquist, B., Osher, S., Chakravarthy, S.R.: Uniformly high order accurate essentially non-oscillatory schemes, III. J. Comput. Phys. 71(2), 231–303 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  7. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  8. Shu, C.-W.: High Order ENO and WENO Schemes for Computational Fluid Dynamics. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  9. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes II. J. Comput. Phys. 83(1), 32–78 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  10. Arandiga, F., Cohen, A., Donat, R., Dyn, N., Matei, B.: Approximation of piecewise smooth functions and images by edge-adapted (ENO-EA) nonlinear multiresolution techniques. Appl. Comput. Harmon. Anal. 24(2), 225–250 (2008). Special Issue on Mathematical Imaging–Part II

    Article  MathSciNet  MATH  Google Scholar 

  11. Amat, S., Aràndiga, F., Cohen, A., Donat, R., Garcia, G., von Oehsen, M.: Data compression with ENO schemes: a case study. Appl. Comput. Harmon. Anal. 11(2), 273–288 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Serna, S., Marquina, A.: Power ENO methods: a fifth-order accurate weighted power ENO method. J. Comput. Phys. 194(2), 632–658 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cohen, A., Dyn, N., Matei, B.: Quasi linear subdivision schemes with applications to ENO interpolation. Appl. Comput. Harmon. Anal. 15, 89–116 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Amat, S., Busquier, S., Trillo, J.C.: On multiresolution schemes using a stencil selection procedure: applications to ENO schemes. Numer. Algorithms 44(1), 45–68 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Liu, X.-D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115(1), 200–212 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  16. Aràndiga, F., Belda, A., Mulet, P.: Point-value WENO multiresolution applications to stable image compression. J. Sci. Comput. 43(2), 158–182 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Henrick, A.K., Aslam, T.D., Powers, J.M.: Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points. J. Comput. Phys. 207(2), 542–567 (2005)

    Article  MATH  Google Scholar 

  18. Castro, M., Costa, B., Don, W.S.: High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws. J. Comput. Phys. 230(5), 1766–1792 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  20. Shu, C.-W.: High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM Rev. 51(1), 82–126 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Borges, R., Carmona, M., Costa, B., Don, W.S.: An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J. Comput. Phys. 227(6), 3191–3211 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Aràndiga, F., Martí, M.C., Mulet, P.: Weights design for maximal order WENO schemes. J. Sci. Comput. 60(3), 641–659 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Yamaleev, N.K., Carpenter, M.H.: A systematic methodology for constructing high-order energy stable WENO schemes. J. Comput. Phys. 228(11), 4248–4272 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Amat, S., Liandrat, J., Ruiz, J., Trillo, J.C.: On a Power-WENO scheme with improved accuracy near discontinuities (in preparation)

  25. Aràndiga, F., Donat, R.: Nonlinear multiscale decompositions: the approach of A. Harten. Numer. Algorithms 23(2–3), 175–216 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. Amat, S., Dadourian, K., Liandrat, J., Ruiz, J., Trillo, J.C.: On a class of \(L^1\)-stable nonlinear cell-average multiresolution schemes. J. Comput. Appl. Math. 234(4), 1129–1139 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Amat, S., Liandrat, J., Ruiz, J., Trillo, J.: On a compact non-extrapolating scheme for adaptive image interpolation. J. Franklin Inst. 349(5), 1637–1647 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Amat, S., Liandrat, J., Ruiz, J., Trillo, J.: On a nonlinear cell-average multiresolution scheme for image compression. SeMA J. 60(1), 75–92 (2012)

    MathSciNet  MATH  Google Scholar 

  29. Amat, S., Ruiz, J., Trillo, J.C.: Adaptive interpolation of images using a new nonlinear cell-average scheme. Math. Comput. Simul. 82(9), 1586–1596 (2012)

    Article  MathSciNet  Google Scholar 

  30. Amat, S., Dadourian, K., Liandrat, J.: Analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms. Adv. Comput. Math. 34(3), 253–277 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Amat, S., Donat, R., Liandrat, J., Trillo, J.: Analysis of a new nonlinear subdivision scheme. Applications in image processing. Found. Comput. Math. 6(2), 193–225 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Amat, S., Ruiz, J., Trillo, J.C.: Improving the compression rate versus \(L^1\) error ratio in cell-average error control algorithms. Numer. Algorithms 67(1), 145–162 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Amat, S., Liandrat, J.: On the stability of the PPH nonlinear multiresolution. Comput. Harmon. Anal. 18(2), 198–206 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  34. Harten, A.: ENO schemes with subcell resolution. J. Comput. Phys. 83(1), 148–184 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  35. Amat, S., Busquier, S., Escudero, A., Trillo, J.C.: Lagrange interpolation for continuous piecewise smooth functions. J. Comput. Appl. Math. 221(1), 47–51 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  36. Amat, S., Dadourian, K., Liandrat, J., Ruiz, J., Trillo, J.C.: A family of stable nonlinear nonseparable multiresolution schemes in 2D. J. Comput. Appl. Math. 234(4), 1277–1290 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zheng, F., Qiu, J.: Directly solving the Hamilton–Jacobi equations by Hermite WENO schemes. J. Comput. Phys. 307, 423–445 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  38. Wu, K., Tang, H.: High-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamics. J. Comput. Phys. 298(C), 539–564 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  39. Huang, C.-S., Xiao, F., Arbogast, T.: Fifth order multi-moment WENO schemes for hyperbolic conservation laws. J. Sci. Comput. 64(2), 477–507 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  40. Hajipour, M., Malek, A.: High accurate modified WENO method for the solution of Black–Scholes equation. Comput. Appl. Math. 34(1), 125–140 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  41. Li, G., Caleffi, V., Gao, J.: High-order well-balanced central WENO scheme for pre-balanced shallow water equations. Comput. Fluids 99, 182–189 (2014)

    Article  MathSciNet  Google Scholar 

  42. Jiang, Y., Xu, Z.: Parametrized maximum principle preserving limiter for finite difference WENO schemes solving convection-dominated diffusion equations. SIAM J. Sci. Comput. 35(6), A2524–A2553 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  43. Kudryavtsev, A.N., Shershnev, A.A.: A numerical method for simulation of microflows by solving directly kinetic equations with WENO schemes. J. Sci. Comput. 57(1), 42–73 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  44. Abedian, R., Adibi, H., Dehghan, M.: A high-order weighted essentially non-oscillatory (WENO) finite difference scheme for nonlinear degenerate parabolic equations. Comput. Phys. Commun. 184(8), 1874–1888 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  45. Gao, Z., Don, W.S.: Mapped hybrid central-WENO finite difference scheme for detonation waves simulations. J. Sci. Comput. 55(2), 351–371 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  46. Deng, W., Du, S., Wu, Y.: High order finite difference WENO schemes for fractional differential equations. Appl. Math. Lett. 26(3), 362–366 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  47. Shen, Y., Zha, G.: Low diffusion E-CUSP scheme with high order WENO scheme for preconditioned Navier–Stokes equations. Cumput. Fluids 55, 13–23 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  48. Qiu, J., Shu, C.-W.: Runge–Kutta discontinuous Galerkin method using WENO limiters. SIAM J. Sci. Comput. , 26(3), 907–929 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics and Statistics, Universidad Politécnica de Cartagena (UPCT), Cartagena, Murcia, Spain

    Sergio Amat

  2. Department of Physics and Mathematics, Universidad de Alcalá (UAH), Alcalá de Henares, Madrid, Spain

    Juan Ruiz

Authors
  1. Sergio Amat
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Juan Ruiz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sergio Amat.

Additional information

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,

$$\begin{aligned} I_1(f)= & {} {\frac{22849}{2880}}\,{{f^2_j}}-{\frac{5423}{160}}\,{ f_j}\,{ f_{j-1}}+{\frac{11489}{480}}\,{ f_j}\,{ f_{j-2}}-{\frac{8509}{1440}} \,{ f_j}\,{ f_{j-3}}\nonumber \\&+{\frac{36107}{960}}\,{{ f^2_{j-1}}}-{\frac{ 26407}{480}}\,{ f_{j-1}}\,{ f_{j-2}} +{\frac{6569}{480}}\,{ f_{j-1}}\,{ f_{j-3}}+{\frac{19907}{960}}\,{{ f^2_{j-2}}}\nonumber \\&-{\frac{1663}{160}}\,{ f_{j-2}} \,{ f_{j-3}}+{\frac{3769}{2880}}\,{{ f^2_{j-3}}} \end{aligned}$$
(64)
$$\begin{aligned} I_2(f)= & {} {\frac{3769}{2880}}\,{{ f^2_{j+1}}}-{\frac{2189}{480}}\,{ f_{j+1}}\,{ f_j}+{\frac{323}{160}}\,{ f_{j+1}}\,{ f_{j-1}}-{\frac{109}{1440}}\,{ f_{j+1}}\,{ f_{j-2}}\nonumber \\&+{\frac{5027}{960}}\,{{ f^2_j}}-{\frac{3127}{ 480}}\,{ f_j}\,{ f_{j-1}} +{\frac{289}{480}}\,{ f_j}\,{ f_{j-2}}+{ \frac{2507}{960}}\,{{ f^2_{j-1}}}\nonumber \\&-{\frac{349}{480}}\,{ f_{j-1}}\,{ f_{j-2}}+{\frac{289}{2880}}\,{{ f^2_{j-2}}} \end{aligned}$$
(65)
$$\begin{aligned} I_3(f)= & {} {\frac{289}{2880}}\,{{ f^2_{j+2}}}-{\frac{349}{480}}\,{ f_{j+2}}\,{ f_{j+1}}+{\frac{289}{480}}\,{ f_{j+2}}\,{ f_j}\nonumber \\&-{\frac{109}{1440}}\,{ f_{j+2}}\,{ f_{j-1}}+{\frac{2507}{960}}\,{{ f^2_{j+1}}}-{\frac{3127}{ 480}}\,{ f_{j+1}}\,{ f_j} +{\frac{323}{160}}\,{ f_{j+1}}\,{ f_{j-1}}+{ \frac{5027}{960}}\,{{ f^2_j}}\nonumber \\&-{\frac{2189}{480}}\,{ f_j}\,{ f_{j-1}}+{\frac{3769}{2880}}\,{{ f^2_{j-1}}}. \end{aligned}$$
(66)

Now we can substitute the Taylor expansions (34) in (64), (65), (66), and we obtain,

$$\begin{aligned} I_1(f)\approx & {} {h}^{2} \left( f'_j \right) ^{2}+ \left( \frac{1}{12}\,{h}^{4}f'''_j -{\frac{15}{4}}\,hO(h^4) \right) f'_j+{\frac{13}{12}} h^{4} \left( f''_j \right) ^{2}\nonumber \\&-\frac{13}{3}\,{h}^{2} \left( f''_j \right) O(h^4)-{\frac{469}{1440}}\,{h}^{3} \left( f'''_j \right) O(h^4) +{\frac{83}{960}}\,{h}^{6} \left( f'''_j \right) ^{2}\nonumber \\&+{\frac{22849}{2880}}\,{O(h^4)}^{2}\approx {h}^{2} \left( f'_j \right) ^{2}\left( 1+O(h^2)\right) \end{aligned}$$
(67)
$$\begin{aligned} I_2(f)\approx & {} {h}^{2} \left( f'_j \right) ^{2}+ \left( \frac{1}{12}\,{h}^{4}f'''_j -\frac{3}{4}\,hO(h^4) \right) f'_j+{\frac{13}{12}}\,{h}^{4} \left( f''_j \right) ^{2}\nonumber \\&+\frac{13}{3}\,{h}^{2} \left( f''_j \right) O(h^4)+{\frac{229}{480}}\,{h}^{3} \left( f'''_j \right) O(h^4)\nonumber \\&+{\frac{83}{960}}\,{h}^{6} \left( f'''_j \right) ^{2}+{\frac{5027}{960}}\,{O(h^4)}^{2}\approx {h}^{2} \left( f'_j \right) ^{2}\left( 1+O(h^2)\right) \end{aligned}$$
(68)
$$\begin{aligned} I_3(f)\approx & {} {h}^{2} \left( f'_j \right) ^{2}+ \left( \frac{1}{12}\,{h}^{4}f'''_j +\frac{3}{4}\,hO(h^4) \right) f'_j+{\frac{13}{12}}\,{h}^{4} \left( f''_j \right) ^{2}\nonumber \\&+\frac{13}{3}\,{h}^{2} \left( f''_j \right) O(h^4)-{\frac{229}{480}}\,{h}^{3} \left( f'''_j \right) O(h^4)\nonumber \\&+{\frac{83}{960}}\,{h}^{6} \left( f'''_j \right) ^{2}+{\frac{5027}{960}}\,{O(h^4)}^{2}\approx {h}^{2} \left( f'_j \right) ^{2}\left( 1+O(h^2)\right) , \end{aligned}$$
(69)

that is the desired result.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-016-0342-2

Keywords

Mathematics Subject Classification

Search

Navigation