Moment-based multi-resolution HWENO scheme for hyperbolic conservation laws. (English) Zbl 1496.65120
Summary: In this paper, a high-order moment-based multi-resolution Hermite weighted essentially non-oscillatory (HWENO) scheme is designed for hyperbolic conservation laws. The main idea of this scheme is derived from our previous work [J. Comput. Phys. 446, Article ID 110653, 21 p. (2021; Zbl 07516463)], in which the integral averages of the function and its first order derivative are used to reconstruct both the function and its first order derivative values at the boundaries. However, in this paper, only the function values at the Gauss-Lobatto points in the one or two dimensional case need to be reconstructed by using the information of the zeroth and first order moments. In addition, an extra modification procedure is used to modify those first order moments in the troubled-cells, which leads to an improvement of stability and an enhancement of resolution near discontinuities. To obtain the same order of accuracy, the size of the stencil required by this moment-based multi-resolution HWENO scheme is still the same as the general HWENO scheme and is more compact than the general WENO scheme. Moreover, the linear weights are not unique and are independent of the node position, and the CFL number can still be 0.6 whether for the one or two dimensional case, which has to be 0.2 in the two dimensional case for other HWENO schemes. Extensive numerical examples are given to demonstrate the stability and resolution of such moment-based multi-resolution HWENO scheme.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 35L65 | Hyperbolic conservation laws |
| 65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
moment-based scheme; multi-resolution scheme; HWENO scheme; hyperbolic conservation laws; KXRCF troubled-cell indicator; HLLC-fluxCitations:
Zbl 07516463References:
| [1] | D.S. Balsara, S. Garain, V. Florinski and W. Boscheri, An efficient class of WENO schemes with adaptive order for unstructured meshes, J. Comput. Phys., 404 (2020), 109062. · Zbl 1453.65208 |
| [2] | D.S. Balsara, S. Garain and C.-W. Shu, An efficient class of WENO schemes with adaptive order, J. Comput. Phys., 326 (2016), 780-804. · Zbl 1422.65146 |
| [3] | D.S. Balsara, C.-W. Shu, Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, J. Comput. Phys., 160 (2000), 405-452. · Zbl 0961.65078 |
| [4] | R. Biswas, K.D. Devine, and J. Flaherty, Parallel, adaptive finite element methods for conser-vation laws, Appl. Numer. Math., 14 (1994), pp. 255-283. · Zbl 0826.65084 |
| [5] | A. Burbeau, P. Sagaut, and C.H. Bruneau, A problem-independent limiter for high-order Runge-Kutta discontinuous Galerkin methods, J. Comput. Phys., 169 (2001), pp. 111-150. · Zbl 0979.65081 |
| [6] | M. Castro, B. Costa, W.S. Don, High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws, J. Comput. Phys., 230 (2011), 1766-1792. · Zbl 1211.65108 |
| [7] | B. Cockburn and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework, Math. Comp., 52 (1989), pp. 411-435. · Zbl 0662.65083 |
| [8] | M. Dumbser, D.S. Balsara, E.F. Toro and C.D. Munz, A unified framework for the construc-tion of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes, J. Comput. Phys., 227 (2008), 8209-8253. · Zbl 1147.65075 |
| [9] | A. Harten, ENO schemes with subcell resolution, J. Comput. Phys., 83 (1989), pp. 148-184. · Zbl 0696.65078 |
| [10] | A. Harten, B. Engquist, S. Osher, S. Chakravarthy, Uniformly high order accurate essentially non-oscillatory schemes, III, J. Comput. Phys., 71 (1987), 231-303. · Zbl 0652.65067 |
| [11] | C. Hu, C.-W. Shu, Weighted essentially non-oscillatory schemes on triangular meshes, J. Comput. Phys., 150 (1999), 97-127. · Zbl 0926.65090 |
| [12] | G.-S. Jiang, C.-W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126 (1996), 202-228. · Zbl 0877.65065 |
| [13] | L. Krivodonova, J. Xin, J.-F. Remacle, N. Chevaugeon, J.E. Flaherty, Shock detection and lim-iting with discontinuous Galerkin methods for hyperbolic conservation laws, Appl. Numer. Math. 48 (2004) 323-338. · Zbl 1038.65096 |
| [14] | J. Li, J. Qiu, C.-W. Shu, Multi-resolution HWENO schemes for hyperbolic conservation laws, J. Comput. Phys., 446 (2021) 110653. · Zbl 07516463 |
| [15] | H. Liu, J. Qiu, Finite difference Hermite WENO schemes for conservation laws, J. Sci. Com-put., 63 (2015), 548-572. · Zbl 1318.65053 |
| [16] | H. Liu, J. Qiu, Finite difference Hermite WENO schemes for conservation laws, II: An alter-native approach, J. Sci. Comput., 66 (2016), 598-624. · Zbl 1398.65215 |
| [17] | X.D. Liu, S. Osher, T. Chan, Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115 (1994), 200-212. · Zbl 0811.65076 |
| [18] | S. Pirozzoli, Conservative hybrid compact-WENO schemes for shock-turbulence interaction, J. Comput. Phys., 178 (2002), 81-117. · Zbl 1045.76029 |
| [19] | J. Qiu, C.-W. Shu, Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: One-dimensional case, J. Comput. Phys., 193 (2004), 115-135. · Zbl 1039.65068 |
| [20] | J. Qiu, C.-W. Shu, A comparison of troubled-cell indicators for Runge-Kutta discontinuous Galerkin methods using weighted essentially nonoscillatory limiters, SIAM J. Sci. Comput. 27 (2005) 995-1013. · Zbl 1092.65084 |
| [21] | W.J. Rider and L.G. Margolin, Simple modifications of monotonicity-preserving limiters, J. Comput. Phys., 174 (2001), pp. 473-488. · Zbl 1009.76067 |
| [22] | C.-W. Shu, S. Osher, Efficient implementation of essentially non-oscillatory shock capturing schemes, J. Comput. Phys., 77 (1988), 439-471. · Zbl 0653.65072 |
| [23] | C.-W. Shu, S. Osher, Efficient implementation of essentially non-oscillatory shock capturing schemes, II, J. Comput. Phys., 83 (1989), 32-78. · Zbl 0674.65061 |
| [24] | A. Suresh and H.T. Huynh, Accurate monotonicity-preserving schemes with Runge-Kutta time stepping, J. Comput. Phys., 136 (1997), pp. 83-99. · Zbl 0886.65099 |
| [25] | Z. Zhao, Y. Chen, J. Qiu, A hybrid Hermite WENO scheme for hyperbolic conservation laws, J. Comput. Phys. 405 (2020) 109175. · Zbl 1453.65264 |
| [26] | Z. Zhao, J. Qiu, A Hermite WENO scheme with artificial linear weights for hyperbolic con-servation laws, J. Comput. Phys., 417 (2020), 109583. · Zbl 1437.76033 |
| [27] | J. Zhu, J. Qiu, A class of fourth order finite volume Hermite weighted essentially non-oscillatory schemes, Sci. China Ser. A, Math., 51 (2008), 1549-1560. · Zbl 1157.65441 |
| [28] | J. Zhu, C.-W. Shu, A new type of multi-resolution WENO schemes with increasingly higher order of accuracy, J. Comput. Phys., 375 (2018), 659-683. · Zbl 1416.65286 |
| [29] | J. Zhu, C.-W. Shu, A new type of multi-resolution WENO schemes with increasingly higher order of accuracy on triangular meshes, J. Comput. Phys., 392 (2019), 19-33. · Zbl 1452.76143 |
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