Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T17:18:20.709Z Has data issue: false hasContentIssue false
  • English
  • Français

A Multi-Domain Hybrid DG and WENO Method for Hyperbolic Conservation Laws on Hybrid Meshes

Published online by Cambridge University Press:  03 June 2015

Jian Cheng  and
Tiegang Liu
Jian Cheng*
Affiliation:
LMIB, School of Mathematics and Systems Science, Beihang University, Beijing 100191, P.R. China
Tiegang Liu*
Affiliation:
LMIB, School of Mathematics and Systems Science, Beihang University, Beijing 100191, P.R. China
*
Email:chengjian@smss.buaa.edu.cn
Corresponding author.Email:liutg@buaa.edu.cn
Get access

Abstract

In [SIAM J. Sci. Comput., 35(2)(2013), A1049-A1072], a class of multi-domain hybrid DG and WENO methods for conservation laws was introduced. Recent applications of this method showed that numerical instability may encounter if the DG flux with Lagrangian interpolation is applied as the interface flux during the moment of conservative coupling. In this continuation paper, we present a more robust approach in the construction of DG flux at the coupling interface by using WENO procedures of reconstruction. Based on this approach, such numerical instability is overcome very well. In addition, the procedure of coupling a DG method with a WENO-FD scheme on hybrid meshes is disclosed in detail. Typical testing cases are employed to demonstrate the accuracy of this approach and the stability under the flexibility of using either WENO-FD flux or DG flux at the moment of requiring conservative coupling.

Keywords

65M6065M9935L65Discontinuous Galerkin methodweighted essentially nonoscillatory schemehybrid methodconservation laws
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 4 , October 2014 , pp. 1116 - 1134
Copyright
Copyright © Global Science Press Limited 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Cockburn, B.Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II: general framework, Math. Comp., 52 (1989), 411–435.Google Scholar
[2]Cockburn, B., Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws III: one dimensional systems, J. Comput. Phys., 84 (1989), 90–113.Google Scholar
[3]Cockburn, B., Shu, C.-W., The Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws V: multidimensional systems, J. Comput. Phys., 141 (1998), 199–224.Google Scholar
[4]Zhu, J., Qiu, J. X., Shu, C.-W., Dumbser, M., Runge-Kutta discontinuous Galerkin method using WENO limiters II: unstructured meshes, J. Comput. Phys., 227 (2008), 4330–4353.Google Scholar
[5]Zhu, J., Qiu, J. X., Hermite WENO schemes and their application as limiter for Runge-Kutta discontinuous Galerkin method III: unstructured meshes, J. Sci. Comput., 39 (2009), 293–321.CrossRefGoogle Scholar
[6]Luo, H., Baum, J. D., Löhner, R., A Hermite WENO-based limiter for discontinuous Galerkin method on unstructed grids, J. Comput. Phys., 225 (2007), 686–713.Google Scholar
[7]Jiang, G. S., Shu, C.-W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126 (1996), 202–228.Google Scholar
[8]Shu, C.-W., Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, NASA/CR-97-206253, ICASE Report No.97–65.Google Scholar
[9]Friedrich, O., Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids, J. Comput. Phys., 144 (1998), 194–212.Google Scholar
[10]Barth, T. J., Frederickson, P. O., High-order solution of the Euler equations on unstructured grids using quadratic reconstruction, AIAA-1990–0013,1990.Google Scholar
[11]Balsara, D. S., Altmann, C., Munz, C. D., Dumbser, M., A sub-cell based indicator for troubled zones in RKDG schemes and a novel class of hybrid RKDG+HWENO schemes, J. Comput. Phys., 226 (2007), 586–620.Google Scholar
[12]Luo, H., Luo, L. Q., Nourgaliev, R., A reconstructed discontinuous Galerkin method for the Euler equations on arbitrary grids., Commun. Comput. Phys., 12 (2012), 1495–1519.CrossRefGoogle Scholar
[13]Luo, H., Xia, Y. D., Nourgaliev, R., Cai, C. P., A class of reconstructed discontinuous Galerkin methods for the compressible flows on arbitrary grids, AIAA-2011–0199,2011.Google Scholar
[14]Dumbser, M., Balsara, D. S., Toro, E. F., A unified framework for the construction of one step finite volume and discontinuous Galerkin schemes on unstructed meshes, J. Comput. Phys., 227 (2008), 8209–8253.Google Scholar
[15]Zhang, L. P., Liu, W., He, L. X., Deng, X. G., Zhang, H. X., A new class of DG/FV hybrid methods for conservation laws I: basic formulation and one-dimensional systems, J. Comput. Phys., 231 (2012), 1081–1103.Google Scholar
[16]Zhang, L. P., Liu, W., He, L. X., Deng, X. G., Zhang, H. X., A class of hybrid DG/FV methods for conservation laws II: two dimensional cases, J. Comput. Phys., 231 (2012), 1104–1120.Google Scholar
[17]Zhang, L. P., Liu, W., He, L. X., Deng, X. G., A class of hybrid DG/FV methods for conservation laws III: two dimensional Euler Equations, Commun. Comput. Phys., 12 (2012), 284–314.Google Scholar
[18]Zhang, L. P., Liu, W., He, L. X., Deng, X. G., A new class of DG/FV hybrid schemes for one-dimensional conservation law, The 8th Asian Conference on Computational Fluid Dynamics, Hong Kong, 10–14 January, 2010.Google Scholar
[19]Wang, Z. J., Spectral (Finite) volume method for conservation laws on unstructured grids: basic formulation, J. Comput. Phys., 178 (2002), 210–251.Google Scholar
[20]Liu, Y., Vinokur, M., Wang, Z. J., Spectral difference method for unstructured grids I: basic formulation, J. Comput. Phys., 216 (2006), 780–801.Google Scholar
[21]Sun, Y., Wang, Z.J. and Liu, Y., High-order multidomain spectral difference method for the Navier-Stokes equations on unstructured hexahedral grids, Commun. Comput. Phys., 2 (2007), 310–333.Google Scholar
[22]Huynh, H. T., A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods, AIAA-2007–4079,2007.Google Scholar
[23]Costa, B., Don, W. S., Multi-domain hybrid spectral-WENO methods for hyperbolic conservation laws, J. Comput. Phys., 224 (2007), 970–991.Google Scholar
[24]Shahbazi, K., Albin, N., Bruno, O. P., Hesthaven, J. S., Multi-domain Fourier-continuation/WENO hybrid solver for conservation laws, J. Comput. Phys., 230 (2011), 8779–8796.Google Scholar
[25]Cheng, J., Lu, Y. W., Liu, T. G., Multi-domain hybrid RKDG and WENO methods for hyperbolic conservation laws, SIAM J. Sci. Comput., 35(2) (2013), A1049–A1072.Google Scholar
[26]Utzmann, J., Lorcher, F., Dumbser, M., Munz, C.-D., Aeroacoustic simulations for complex geometries based on hybrid meshes, AIAA-2006–2418,2006.Google Scholar
[27]Utzmann, J., Schwartzkopff, T., Dumbser, M., Munz, C.-D., Heterogeneous domain decomposition for computational aeroacoustics, AIAA jouirnal, 44(10) (2006), 2231–2250.Google Scholar
[28]Léger, R., Peyret, C., Piperno, S., Coupled discontinuous Galerkin/finite difference solver on hybrid meshes for computational aeroacoutics., AIAA journal, 50(2) (2012), 338–349.Google Scholar