# zbMATH — the first resource for mathematics

CWENO: uniformly accurate reconstructions for balance laws. (English) Zbl 1412.65102
Summary: In this paper we introduce a general framework for defining and studying essentially nonoscillatory reconstruction procedures of arbitrarily high order of accuracy, interpolating data in the central stencil around a given computational cell ($$\mathsf {CWENO}$$). This technique relies on the same selection mechanism of smooth stencils adopted in $$\mathsf {WENO}$$, but here the pool of candidates for the selection includes polynomials of different degrees. This seemingly minor difference allows us to compute the analytic expression of a polynomial interpolant, approximating the unknown function uniformly within a cell, instead of only at one point at a time. For this reason this technique is particularly suited for balance laws for finite volume schemes, when averages of source terms require high order quadrature rules based on several points; in the computation of local averages, during refinement in $$h$$-adaptive schemes; or in the transfer of the solution between grids in moving mesh techniques, and in general when a globally defined reconstruction is needed. Previously, these needs have been satisfied mostly by ENO reconstruction techniques, which, however, require a much wider stencil than the $$\mathsf {CWENO}$$ reconstruction studied here, for the same accuracy.

##### MSC:
 65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
SharpClaw
Full Text:
##### References:
 [1] Ar\`andiga, 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) · Zbl 1233.65051 [2] Audusse, Emmanuel; Bouchut, Fran\ccois; Bristeau, Marie-Odile; Klein, Rupert; Perthame, Beno\^\i t., A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sci. Comput., 25, 6, 2050-2065, (2004) · Zbl 1133.65308 [3] \bibButcher:2008book author=Butcher, J. C., title=Numerical Methods for Ordinary Differential Equations, edition=2, pages=xx+463, publisher=John Wiley & Sons, Ltd., Chichester, date=2008, doi=10.1002/9780470753767, isbn=978-0-470-72335-7, review=\MR2401398, [4] Capdeville, G., A central WENO scheme for solving hyperbolic conservation laws on non-uniform meshes, J. Comput. Phys., 227, 5, 2977-3014, (2008) · Zbl 1135.65359 [5] Carlini, E.; Ferretti, R.; Russo, G., A weighted essentially nonoscillatory, large time-step scheme for Hamilton-Jacobi equations, SIAM J. Sci. Comput., 27, 3, 1071-1091, (2005) · Zbl 1105.65090 [6] Cravero, I.; Semplice, M., On the accuracy of WENO and CWENO reconstructions of third order on nonuniform meshes, J. Sci. Comput., 67, 3, 1219-1246, (2016) · Zbl 1343.65116 [7] Curtis, A. R., High-order explicit Runge-Kutta formulae, their uses, and limitations, J. Inst. Math. Appl., 16, 1, 35-55, (1975) · Zbl 0317.65024 [8] Don, Wai-Sun; Borges, Rafael, Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes, J. Comput. Phys., 250, 347-372, (2013) · Zbl 1349.65285 [9] Dumbser, Michael; Balsara, Dinshaw S.; Toro, Eleuterio F.; Munz, Claus-Dieter, A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes, J. Comput. Phys., 227, 18, 8209-8253, (2008) · Zbl 1147.65075 [10] Dumbser, Michael; K\"aser, Martin, Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems, J. Comput. Phys., 221, 2, 693-723, (2007) · Zbl 1110.65077 [11] Feng, Hui; Huang, Cong; Wang, Rong, An improved mapped weighted essentially non-oscillatory scheme, Appl. Math. Comput., 232, 453-468, (2014) · Zbl 1410.65306 [12] Gerolymos, G. A., Representation of the Lagrange reconstructing polynomial by combination of substencils, J. Comput. Appl. Math., 236, 11, 2763-2794, (2012) · Zbl 06031322 [13] Gorsse, Yannick; Iollo, Angelo; Telib, Haysam; Weynans, Lisl, A simple second order Cartesian scheme for compressible Euler flows, J. Comput. Phys., 231, 23, 7780-7794, (2012) · Zbl 1255.76081 [14] Ha, Youngsoo; Kim, Chang Ho; Lee, Yeon Ju; Yoon, Jungho, An improved weighted essentially non-oscillatory scheme with a new smoothness indicator, J. Comput. Phys., 232, 68-86, (2013) · Zbl 1291.65264 [15] HOEC:1986 A. Harten, S. Osher, B. Engquist, and S. Chachravarty, \emph Uniformly high order accurate essentially non-oscillatory schemes III, 1986, NASA ICASE report 86-22. [16] HAP:2005:mappedWENO A. K. Henrick, T. D. Aslam, and J. M. Powers, \emph Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points, J. Comput. Phys. 207 (2005), 542–567. · Zbl 1072.65114 [17] Hu, Changqing; Shu, Chi-Wang, Weighted essentially non-oscillatory schemes on triangular meshes, J. Comput. Phys., 150, 1, 97-127, (1999) · Zbl 0926.65090 [18] Jiang, Guang-Shan; Shu, Chi-Wang, Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126, 1, 202-228, (1996) · Zbl 0877.65065 [19] Ketcheson, David I.; Parsani, Matteo; LeVeque, Randall J., High-order wave propagation algorithms for hyperbolic systems, SIAM J. Sci. Comput., 35, 1, A351-A377, (2013) · Zbl 1264.65151 [20] Kolb, Oliver, On the full and global accuracy of a compact third order WENO scheme, SIAM J. Numer. Anal., 52, 5, 2335-2355, (2014) · Zbl 1408.65062 [21] Kurganov, Alexander; Petrova, Guergana, A second-order well-balanced positivity preserving central-upwind scheme for the Saint-Venant system, Commun. Math. Sci., 5, 1, 133-160, (2007) · Zbl 1226.76008 [22] Levy, Doron; Puppo, Gabriella; Russo, Giovanni, Central WENO schemes for hyperbolic systems of conservation laws, M2AN Math. Model. Numer. Anal., 33, 3, 547-571, (1999) · Zbl 0938.65110 [23] Levy, Doron; Puppo, Gabriella; Russo, Giovanni, Compact central WENO schemes for multidimensional conservation laws, SIAM J. Sci. Comput., 22, 2, 656-672, (2000) · Zbl 0967.65089 [24] Noelle, Sebastian; Pankratz, Normann; Puppo, Gabriella; Natvig, Jostein R., Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows, J. Comput. Phys., 213, 2, 474-499, (2006) · Zbl 1088.76037 [25] Puppo, Gabriella, Adaptive application of characteristic projection for central schemes. Hyperbolic Problems: Theory, Numerics, Applications, 819-829, (2003), Springer, Berlin · Zbl 1064.65106 [26] Puppo, Gabriella, Numerical entropy production for central schemes, SIAM J. Sci. Comput., 25, 4, 1382-1415, (2003/04) · Zbl 1061.65094 [27] Puppo, Gabriella; Semplice, Matteo, Numerical entropy and adaptivity for finite volume schemes, Commun. Comput. Phys., 10, 5, 1132-1160, (2011) · Zbl 1373.76140 [28] PS:HYP12 Gabriella Puppo and Matteo Semplice, \emph Finite Volume Schemes on 2d Non-uniform Grids, Proceedings of “Fourteenth International Conference devoted to Theory, Numerics and Applications of Hyperbolic Problems” (HYP2012) (AIMS, ed.), 2014. [29] Puppo, G.; Semplice, M., Well-balanced high order 1D schemes on non-uniform grids and entropy residuals, J. Sci. Comput., 66, 3, 1052-1076, (2016) · Zbl 1371.65093 [30] Qiu, Jianxian; Shu, Chi-Wang, On the construction, comparison, and local characteristic decomposition for high-order central WENO schemes, J. Comput. Phys., 183, 1, 187-209, (2002) · Zbl 1018.65106 [31] Semplice, M.; Coco, A.; Russo, G., Adaptive mesh refinement for hyperbolic systems based on third-order compact WENO reconstruction, J. Sci. Comput., 66, 2, 692-724, (2016) · Zbl 1335.65077 [32] ShiHuShu:2002 J. Shi, C. Hu, and C.-W. Shu, \emph A technique of treating negative weights in WENO schemes, J. Comput. Phys. 175 (2002), no. 1, 108–127. · Zbl 0992.65094 [33] Shu, Chi-Wang, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Cetraro, 1997, Lecture Notes in Math. 1697, 325-432, (1998), Springer, Berlin · Zbl 0927.65111 [34] Shu, Chi-Wang, High order weighted essentially nonoscillatory schemes for convection dominated problems, SIAM Rev., 51, 1, 82-126, (2009) · Zbl 1160.65330 [35] Tang, Huazhong; Tang, Tao, Adaptive mesh methods for one- and two-dimensional hyperbolic conservation laws, SIAM J. Numer. Anal., 41, 2, 487-515, (2003) · Zbl 1052.65079 [36] Toro, Eleuterio F., Riemann Solvers and Numerical Methods for Fluid Dynamics, xxiv+724 pp., (2009), Springer-Verlag, Berlin · Zbl 1227.76006 [37] Wang, Rong; Feng, Hui; Spiteri, Raymond J., Observations on the fifth-order WENO method with non-uniform meshes, Appl. Math. Comput., 196, 1, 433-447, (2008) · Zbl 1134.65060 [38] Xing, Yulong; Shu, Chi-Wang, High order finite difference WENO schemes with the exact conservation property for the shallow water equations, J. Comput. Phys., 208, 1, 206-227, (2005) · Zbl 1114.76340
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.