High-order multi-resolution central Hermite WENO schemes for hyperbolic conservation laws. (English) Zbl 07835089
Summary: In this paper, a class of high-order multi-resolution central Hermite WENO (C-HWENO) schemes for solving hyperbolic conservation laws is proposed. Formulated in a central finite volume framework on staggered meshes, the methods adopt the multi-resolution HWENO reconstructions (Li et al. in J Comput Phys 446:110653, 2021; Li et al. in Commun Comput Phys 32(2): 364-400, 2022) in space and the natural continuous extension of Runge-Kutta methods in time. Based on the zeroth-order and first-order moments of the solution defined on a series of hierarchical central spatial stencils, the proposed methods are sixth-order while the C-HWENO methods by Tao et al. (J Comput Phys 318:222-251, 2016) are fifth-order in accuracy. The linear weights of such HWENO reconstructions can be any positive numbers as long as their sum equals one, which leads to much simpler implementation and better cost efficiency than the methods by Tao et al. (J Comput Phys 318:222-251, 2016). The first-order moments are modified and the HWENO reconstructions are applied in the troubled-cells, while the linear reconstructions are used for the rest. Meanwhile, our new methods have compact stencils in the reconstructions and require neither numerical fluxes nor flux splitting. Extensive one- and two-dimensional numerical examples are performed to illustrate the accuracy and high resolution of the new C-HWENO schemes.
MSC:
| 65Mxx | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 35Lxx | Hyperbolic equations and hyperbolic systems |
| 76Mxx | Basic methods in fluid mechanics |
Keywords:
finite volume method; central scheme; multi-resolution Hermite WENO; natural continuous extension (NCE) of Runge-KuttaReferences:
| [1] | Balsara, DS; Shu, C-W, Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, J. Comput. Phys., 160, 2, 405-452, 2000 · Zbl 0961.65078 · doi:10.1006/jcph.2000.6443 |
| [2] | Bianco, F.; Puppo, G.; Russo, G., High-order central schemes for hyperbolic systems of conservation laws, SIAM J. Sci. Comput., 21, 1, 294-322, 1999 · Zbl 0940.65093 · doi:10.1137/S1064827597324998 |
| [3] | Borges, R.; Carmona, M.; Costa, B.; Don, WS, An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws, J. Comput. Phys., 227, 6, 3191-3211, 2008 · Zbl 1136.65076 · doi:10.1016/j.jcp.2007.11.038 |
| [4] | Castro, M.; Costa, B.; Don, WS, High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws, J. Comput. Phys., 230, 5, 1766-1792, 2011 · Zbl 1211.65108 · doi:10.1016/j.jcp.2010.11.028 |
| [5] | Dumbser, M.; Boscheri, W.; Semplice, M.; Russo, G., Central WENO schemes for hyperbolic conservation laws on fixed and moving unstructured meshes, SIAM J. Sci. Comput., 39, 6, 2564, 2016 · Zbl 1377.65115 · doi:10.1137/17M1111036 |
| [6] | Hu, C.; Shu, C-W, Weighted essentially non-oscillatory schemes on triangular meshes, J. Comput. Phys., 150, 1, 97-127, 1999 · Zbl 0926.65090 · doi:10.1006/jcph.1998.6165 |
| [7] | Jiang, GS; Shu, C-W, Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126, 1, 202-228, 1996 · Zbl 0877.65065 · doi:10.1006/jcph.1996.0130 |
| [8] | Jiang, G-S; Tadmor, E., Nonoscillatory central schemes for multidimensional hyperbolic conservation laws, SIAM J. Sci. Comput., 19, 6, 1892-1917, 1998 · Zbl 0914.65095 · doi:10.1137/S106482759631041X |
| [9] | Krivodonova, L.; Xin, J.; Remacle, JF; Chevaugeon, N.; Flaherty, JE, Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws, Appl. Numer. Math., 48, 3, 323-338, 2004 · Zbl 1038.65096 · doi:10.1016/j.apnum.2003.11.002 |
| [10] | Kurganov, A.; Levy, D., A third-order semidiscrete central scheme for conservation laws and convection-diffusion equations, SIAM J. Sci. Comput., 22, 4, 1461-1488, 2000 · Zbl 0979.65077 · doi:10.1137/S1064827599360236 |
| [11] | Levy, D.; Puppo, G.; Russo, G., Central WENO schemes for hyperbolic systems of conservation laws, Esaim Math. Model. Numer. Anal., 33, 3, 547-571, 1999 · Zbl 0938.65110 · doi:10.1051/m2an:1999152 |
| [12] | Levy, D.; Puppo, G.; Russo, G., A fourth-order central WENO scheme for multidimensional hyperbolic systems of conservation laws, SIAM J. Sci. Comput., 24, 2, 480-506, 2002 · Zbl 1014.65079 · doi:10.1137/S1064827501385852 |
| [13] | Li, J.; Shu, C-W; Qiu, J., Multi-resolution HWENO schemes for hyperbolic conservation laws, J. Comput. Phys., 446, 2021 · Zbl 07516463 · doi:10.1016/j.jcp.2021.110653 |
| [14] | Li, J.; Shu, C-W; Qiu, J., Moment-based multi-resolution HWENO scheme for hyperbolic conservation laws, Commun. Comput. Phys., 32, 2, 364-400, 2022 · Zbl 1496.65120 · doi:10.4208/cicp.OA-2022-0030 |
| [15] | Liu, H.; Qiu, J., Finite difference Hermite WENO schemes for conservation laws, J. Sci. Comput., 63, 548-572, 2015 · Zbl 1318.65053 · doi:10.1007/s10915-014-9905-2 |
| [16] | Liu, XD; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115, 1, 200-212, 1994 · Zbl 0811.65076 · doi:10.1006/jcph.1994.1187 |
| [17] | Liu, X-D; Tadmor, E., Third order nonoscillatory central scheme for hyperbolic conservation laws, Numer. Math., 79, 3, 397-425, 1998 · Zbl 0906.65093 · doi:10.1007/s002110050345 |
| [18] | Liu, Y., Central schemes on overlapping cells, J. Comput. Phys., 209, 1, 82-104, 2005 · Zbl 1076.65076 · doi:10.1016/j.jcp.2005.03.014 |
| [19] | Nessyahu, H.; Tadmor, E., Non-oscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys., 87, 408-463, 1990 · Zbl 0697.65068 · doi:10.1016/0021-9991(90)90260-8 |
| [20] | Qiu, J.; Shu, C-W, 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 · doi:10.1006/jcph.2002.7191 |
| [21] | Qiu, J.; Shu, C-W, Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: one-dimensional case, J. Comput. Phys., 193, 1, 115-135, 2004 · Zbl 1039.65068 · doi:10.1016/j.jcp.2003.07.026 |
| [22] | Qiu, J.; Shu, C-W, A comparison of troubled-cell indicators for Runge-Kutta discontinuous Galerkin methods using weighted essentially nonoscillatory limiters, SIAM J. Sci. Comput., 27, 3, 995-1013, 2005 · Zbl 1092.65084 · doi:10.1137/04061372X |
| [23] | Qiu, J.; Shu, C-W, Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method II: two dimensional case, Comput. Fluids, 34, 642-663, 2005 · Zbl 1134.65358 · doi:10.1016/j.compfluid.2004.05.005 |
| [24] | Shi, J.; Hu, C.; Shu, C-W, A technique of treating negative weights in WENO schemes, J. Comput. Phys., 175, 1, 108-127, 2002 · Zbl 0992.65094 · doi:10.1006/jcph.2001.6892 |
| [25] | Shu, C-W, High order weighted essentially nonoscillatory schemes for convection dominated problems, SIAM Rev., 51, 1, 82-126, 2009 · Zbl 1160.65330 · doi:10.1137/070679065 |
| [26] | Tao, Z.; Li, F.; Qiu, J., High-order central Hermite WENO schemes on staggered meshes for hyperbolic conservation laws, J. Comput. Phys., 281, 148-176, 2015 · Zbl 1352.65271 · doi:10.1016/j.jcp.2014.10.027 |
| [27] | Tao, Z.; Li, F.; Qiu, J., High-order central Hermite WENO schemes: dimension-by-dimension moment-based reconstructions, J. Comput. Phys., 318, 222-251, 2016 · Zbl 1349.65385 · doi:10.1016/j.jcp.2016.05.005 |
| [28] | Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54, 115-173, 1984 · Zbl 0573.76057 · doi:10.1016/0021-9991(84)90142-6 |
| [29] | Zahran, YH; Abdalla, AH, Seventh order Hermite WENO scheme for hyperbolic conservation laws, Comput. Fluids, 131, 66-80, 2016 · Zbl 1390.76641 · doi:10.1016/j.compfluid.2016.03.010 |
| [30] | Zennaro, M., Natural continuous extensions of Runge-Kutta methods, Math. Comput., 46, 119-133, 1986 · Zbl 0608.65043 · doi:10.1090/S0025-5718-1986-0815835-1 |
| [31] | Zhao, Z.; Chen, Y.; Qiu, J., A hybrid Hermite WENO scheme for hyperbolic conservation laws, J. Comput. Phys., 405, 2020 · Zbl 1453.65264 · doi:10.1016/j.jcp.2019.109175 |
| [32] | Zhao, Z.; Qiu, J., A Hermite WENO scheme with artificial linear weights for hyperbolic conservation laws, J. Comput. Phys., 417, 2020 · Zbl 1437.76033 · doi:10.1016/j.jcp.2020.109583 |
| [33] | Zhu, J.; Qiu, J., A class of the fourth order finite volume Hermite weighted essentially non-oscillatory schemes, Sci. China, 51, 8, 1549-1560, 2008 · Zbl 1157.65441 · doi:10.1007/s11425-008-0105-0 |
| [34] | Zhu, J.; Shu, C-W, A new type of multi-resolution WENO schemes with increasingly higher order of accuracy, J. Comput. Phys., 375, 659-683, 2018 · Zbl 1416.65286 · doi:10.1016/j.jcp.2018.09.003 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
