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Runge-Kutta Discontinuous Galerkin Method Using Weno-Type Limiters: Three-Dimensional Unstructured Meshes

Published online by Cambridge University Press:  20 August 2015

Jun Zhu  and
Jianxian Qiu
Jun Zhu*
Affiliation:
College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 210016, P.R. China
Jianxian Qiu*
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, P.R. China and Department of Mathematics, Nanjing University, Nanjing, Jiangsu 210093, P.R. China
*
Corresponding author.Email:[email protected], [email protected]
Email:[email protected]
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Abstract

This paper further considers weighted essentially non-oscillatory (WENO) and Hermite weighted essentially non-oscillatory (HWENO) finite volume methods as limiters for Runge-Kutta discontinuous Galerkin (RKDG) methods to solve problems involving nonlinear hyperbolic conservation laws. The application discussed here is the solution of 3-D problems on unstructured meshes. Our numerical tests again demonstrate this is a robust and high order limiting procedure, which simultaneously achieves high order accuracy and sharp non-oscillatory shock transitions.

Keywords

65M0665M9935L65Runge-Kutta discontinuous Galerkin methodlimiterWENOHWENOhigh order limiting procedure
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 3 , March 2012 , pp. 985 - 1005
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Biswas, R., Devine, K.D. and Flaherty, J., Parallel, adaptive finite element methods for conservation laws, Applied Numerical Mathematics, 14 (1994), 255–283.Google Scholar
[2]Burbeau, A., Sagaut, P. and Bruneau, C.H., A problem-independent limiter for high-order Runge-Kutta discontinuous Galerkin methods, Journal of Computational Physics, 169 (2001), 111–150.CrossRefGoogle Scholar
[3]Cockburn, B., Hou, S. and Shu, C.-W., The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case, Mathematics of Computation, 54 (1990), 545–581.Google Scholar
[4]Cockburn, B., Lin, S.-Y. and Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One dimensional systems, Journal of Computational Physics, 84 (1989), 90–113.Google Scholar
[5]Cockburn, B. and Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Mathematics of Computation, 52 (1989), 411–435.Google Scholar
[6]Cockburn, B. and Shu, C.-W., The Runge-Kutta local projection P1-discontinuous Galerkin finite element method for scalar conservation laws, Mathematical Modelling and Numerical Analysis (M2AN), 25 (1991), 337–361.Google Scholar
[7]Cockburn, B. and Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems, Journal of Computational Physics, 141 (1998), 199–224.CrossRefGoogle Scholar
[8]Cockburn, B. and Shu, C.-W., Runge-Kutta discontinuous Galerkin method for convection-dominated problems, Journal of Scientific Computing, 16 (2001), 173–261.CrossRefGoogle Scholar
[9]Friedrichs, O., Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids, Journal of Computational Physics, 144 (1998), 194–212.CrossRefGoogle Scholar
[10]Harten, A., Engquist, B., Osher, S. and Chakravathy, S., Uniformly high order accurate essentially non-oscillatory schemes, III, Journal of Computational Physics, 71 (1987), 231–303.Google Scholar
[11]Hu, C. and Shu, C.-W., Weighted essentially non-oscillatory schemes on triangular meshes, Journal of Computational Physics, 150 (1999), 97–127.Google Scholar
[12]Jiang, G. and Shu, C.-W., Efficient implementation of weighted ENO schemes, Journal of Computational Physics, 126 (1996), 202–228.Google Scholar
[13]Kroll, N.et al. (Editors), ADIGMA – A European Initiative on the Development of Adaptive Higher-Order Variational Methods for Aerospace, Applications Notes on Numerical Fluid Mechanics and Multidisciplinary Design, V113, 2010, Springer.Google Scholar
[14]Liu, X., Osher, S. and Chan, T., Weighted essentially non-oscillatory schemes, Journal of Com-putational Physics, 115 (1994), 200–212.CrossRefGoogle Scholar
[15]Qiu, J. and Shu, C.-W., On the construction, comparison, and local characteristic decomposition for high order central WENO schemes, Journal of Computational Physics, 183 (2002), 187–209.CrossRefGoogle Scholar
[16]Qiu, J. and Shu, C.-W., Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: One dimensional case, Journal of Computational Physics, 193 (2003), 115–135.Google Scholar
[17]Qiu, J. and Shu, C.-W., Runge-Kutta discontinuous Galerkin method using WENO limiters, SIAM Journal on Scientific Computing, 26 (2005), 907–929.Google Scholar
[18]Qiu, J. and Shu, C.-W., Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method II: Two dimensional case, Computers & Fluids, 34 (2005), 642–663.CrossRefGoogle Scholar
[19]Reed, W.H. and Hill, T.R., Triangular mesh methods for neutron transport equation, Tech. Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973.Google Scholar
[20]Schmitt, V. and Charpin, F., Pressure Distributions on the ONERA-M6-Wing at Transonic Mach Numbers, Experimental Data Base for Computer Program Assessment. Report of the Fluid Dynamics Panel Working Group 04, AGARD AR 138, May 1979.Google Scholar
[21]Shi, J., Hu, C. and Shu, C.-W., A technique of treating negative weights in WENO schemes, Journal of Computational Physics, 175 (2002), 108–127.Google Scholar
[22]Shu, C.-W., TVB uniformly high-order schemes for conservation laws, Mathematics of Com-putation, 49 (1987), 105–121.Google Scholar
[23]Shu, C.-W. and Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, Journal of Computational Physics, 77 (1988), 439–471.Google Scholar
[24]Shu, C.-W. and Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes II, Journal of Computational Physics, 83 (1989), 32–78.Google Scholar
[25]Zhang, Y.T. and Shu, C.-W., Third order WENO scheme on three dimensional tetrahedral meshes, Communications in Computational Physics, 5 (2009), 836–848.Google Scholar
[26]Zhu, J. and Qiu, J., A class of the fourth order finite volume Hermite weighted essentially non-oscillatory schemes, Science in China, Series A-Mathematics, 51 (2008), 1549–1560.Google Scholar
[27]Zhu, J. and Qiu, J., Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method III: Unstructured meshes, Journal of Scientific Computing, 39 (2009), 293–321.CrossRefGoogle Scholar
[28]Zhu, J., Qiu, J., Shu, C.-W. and Dumbser, M., Runge-Kutta discontinuous Galerkin method using WENO limiters II: Unstructured meshes, Journal of Computational Physics, 227 (2008), 4330–4353.Google Scholar