Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-28T01:12:29.809Z Has data issue: false hasContentIssue false
  • English
  • Français

A Third Order Adaptive ADER Scheme for One Dimensional Conservation Laws

Part of: Hyperbolic equations and systems Equations of mathematical physics and other areas of application Partial differential equations, initial value and time-dependent initial-boundary value problems Partial differential equations on manifolds; differential operators

Published online by Cambridge University Press:  06 July 2017

Yaguang Gu  and
Guanghui Hu
Yaguang Gu*
Affiliation:
Department of Mathematics, University of Macau, Macao SAR, China
Guanghui Hu*
Affiliation:
Department of Mathematics, University of Macau, Macao SAR, China UM Zhuhai Research Institute, Zhuhai, Guangdong, China
*
*Corresponding author. Email addresses:[email protected] (Y. G. Gu), [email protected] (G. H. Hu)
*Corresponding author. Email addresses:[email protected] (Y. G. Gu), [email protected] (G. H. Hu)
Get access

Abstract

We introduce a third order adaptive mesh method to arbitrary high order Godunov approach. Our adaptive mesh method consists of two parts, i.e., mesh-redistribution algorithm and solution algorithm. The mesh-redistribution algorithm is derived based on variational approach, while a new solution algorithm is developed to preserve high order numerical accuracy well. The feature of proposed Adaptive ADER scheme includes that 1). all simulations in this paper are stable for large CFL number, 2). third order convergence of the numerical solutions is successfully observed with adaptive mesh method, and 3). high resolution and non-oscillatory numerical solutions are obtained successfully when there are shocks in the solution. A variety of numerical examples show the feature well.

Keywords

ADER schemeadaptive mesh methodfinite volume methodWENO reconstructionhyperbolic equations

MSC classification

Secondary: 35L65: Conservation laws 35L67: Shocks and singularities 58J45: Hyperbolic equations 35Q31: Euler equations 65M08: Finite volume methods
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 3 , September 2017 , pp. 829 - 851
Copyright
Copyright © Global-Science Press 2017 

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] Fryxell, B., http://cococubed.asu.edu/codes/riemann/exact_riemann.f.Google Scholar
[2] Borges, R., Carmona, M., Costa, B., and Don, W. S., An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws, J. Comput. Phys., 227(6):3191–3211, 2008.Google Scholar
[3] Cao, W.M., Huang, W. Z. and Russell, R. D., A study ofmonitor functions for two-dimensional adaptive mesh generation, SIAM J. Sci. Comput., 20(6):1978–1994, 1999.Google Scholar
[4] Cheng, J. Bo., Toro, E. F., Jiang, S., and Tang, W. J., A sub-cell WENO reconstruction method for spatial derivatives in the ADER scheme, J. Comput. Phys., 251:53–80, 2013.CrossRefGoogle Scholar
[5] Ciarlet, P. G., The finite element method for elliptic problems, volume 40, SIAM, 2002.Google Scholar
[6] Cockburn, B., Dong, B., Guzmán, J., Restelli, M., and Sacco, R., A hybridizable discontinuous Galerkin method for steady-state convection-diffusion-reaction problems, SIAM J. Sci. Comput., 31(5):3827–3846, 2009.Google Scholar
[7] Cockburn, B. and Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework, Math. Comp., 52(186):411–435, 1989.Google Scholar
[8] Cockburn, B. and Zhang, W. J., A posteriori error analysis for hybridizable discontinuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal., 51(1):676–693, 2013.Google Scholar
[9] Gelb, A. and Tadmor, E., Adaptive edge detectors for piecewise smooth data based on the minmod limiter, J. Sci. Comput., 28(2-3):279–306, 2006.Google Scholar
[10] Gottlieb, S. and Shu, C.-W., Total variation diminishing Runge-Kutta schemes, Math. Comp., 67(221):73–85, 1998.Google Scholar
[11] Hu, F. X., Wang, R., Chen, X. Y., and Feng, H., An adaptive mesh method for 1D hyperbolic conservation laws, Appl. Numer. Math., 91:11–25, 2015.Google Scholar
[12] Hu, G. H., An adaptive finite volume method for 2D steady Euler equations with WENO reconstruction, J. Comput. Phys., 252:591–605, 2013.Google Scholar
[13] Hu, G. H., Li, R., and Tang, T., A robust high-order residual distribution type scheme for steady Euler equations on unstructured grids, J. Comput. Phys., 229(5):1681–1697, 2010.CrossRefGoogle Scholar
[14] Hu, G. H., Li, R., and Tang, T., A robust WENO type finite volume solver for steady Euler equations on unstructured grids, Commun. Comput. Phys., 9(3):627–648, 2011.Google Scholar
[15] Huang, W. Z. and Russell, R. D., Moving mesh strategy based on a gradient flow equation for two-dimensional problems, SIAM J. Sci. Comput., 20(3):998–1015, 1998.Google Scholar
[16] Ji, X., Tang, H. Z., High-order accurate Runge-Kutta (local) discontinuous Galerkin methods for one-and two-dimensional fractional diffusion equations, Numer. Math. Theor. Meth. Appl., 5(3):333–358, 2012.Google Scholar
[17] Jiang, G.-S. and Shu, C.-W., Efficient implementation of weighted ENO schemes, Technical report, DTIC Document, 1995.Google Scholar
[18] Kannan, R. and Wang, Z. J., Improving the high order spectral volume formulation using a diffusion regulator, Commun. Comput. Phys., 12(01):247–260, 2012.CrossRefGoogle Scholar
[19] Kannan, R. and Wang, Z. J., A high order spectral volume solution to the Burgers’ equation using the Hopf–Cole transformation, Internat. J. Numer. Methods Fluids, 69(4):781–801, 2012.Google Scholar
[20] Krivodonova, L., Limiters for high-order discontinuous Galerkin methods, J. Comput. Phys., 226(1):879–896, 2007.Google Scholar
[21] LeVeque, R. J., Finite volume methods for hyperbolic problems, volume 31, Cambridge Univ. Press, 2002.Google Scholar
[22] LeVeque, R. J., Numerical methods for conservation laws, volume 132, Birkhäuser Basel, 1992.Google Scholar
[23] Li, R., Tang, T., and Zhang, P. W., Moving mesh methods in multiple dimensions based on harmonic maps, J. Comput. Phys., 170(2):562–588, 2001.Google Scholar
[24] Liu, J., Goman, M., Li, X., and Liu, M., Positivity-preserving Runge-Kutta discontinuous Galerkin method on adaptive Cartesian grid for strong moving shock, Numer. Math. Theor. Meth. Appl., 9(1):87–110, 2016.Google Scholar
[25] Millington, R. C., Titarev, V. A., and Toro, E. F., Freistühler, H., Warnecke, G. (Eds.) ADER: Arbitrary order non-oscillatory advection schemes hyperbolic problems: Theory, Numerics, Applications, volume 141:723–732, Birkhäuser Basel, 2001.Google Scholar
[26] Nguyen, N. C., Peraire, J., and Cockburn, B., A class of embedded discontinuous Galerkin methods for computational fluid dynamics, J. Comput. Phys., 302:674–692, 2015.Google Scholar
[27] Qiu, J. X. and 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.Google Scholar
[28] Remski, J., Mesh spacing estimates and efficiency considerations for moving mesh systems, Numer. Math. Theor. Meth. Appl., 9(3):432–450, 2016.Google Scholar
[29] Shu, C.-W., Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, Springer, 1998.Google Scholar
[30] Tang, H. Z. and Tang, T., Adaptive mesh methods for one-and two-dimensional hyperbolic conservation laws, SIAM J. Numer. Anal., 41(2):487–515, 2003.Google Scholar
[31] Titarev, V. A. and Toro, E. F., ADER schemes for three-dimensional non-linear hyperbolic systems, J. Comput. Phys., 204(2):715–736, 2005.Google Scholar
[32] Titarev, V. A. and Toro, E. F., ADER: Arbitrary high order Godunov approach, J. Sci. Comput., 17(1-4):609–618, 2002.Google Scholar
[33] Toro, E. F., Riemann solvers and numerical methods for fluid dynamics: a practical introduction, Springer, 2009.Google Scholar
[34] Toro, E. F. and Titarev, V. A., Derivative Riemann solvers for systems of conservation laws and ADER methods, J. Comput. Phys., 212(1):150–165, 2006.Google Scholar
[35] Toro, E. F. and Titarev, V. A., Solution of the generalized Riemann problem for advection–reaction equations, Proc. Roy. Soc. London A, volume 458:271–281, 2002.Google Scholar
[36] Wang, C., Zhang, X. X., Shu, C.-W., and Ning, Jianguo, Robust high order discontinuous Galerkin schemes for two-dimensional gaseous detonations, J. Comput. Phys., 231(2):653–665, 2012.Google Scholar
[37] Wang, Z. J., Fidkowski, K., Abgrall, R., Bassi, F., Caraeni, D., Cary, A., Deconinck, H., Hartmann, R., Hillewaert, K., Huynh, H. T., et al, High-order CFD methods: current status and perspective, Internat. J. Numer. Methods Fluids, 72(8):811–845, 2013.Google Scholar
[38] Wu, K. L. and Tang, H. Z., Finite volume local evolution Galerkin method for two-dimensional relativistic hydrodynamics, J. Comput. Phys., 256:277–307, 2014.Google Scholar
[39] Zhang, Q. and Shu, C.-W., Error estimates for the third order explicit Runge-Kutta discontinuous Galerkin method for a linear hyperbolic equation in one-dimension with discontinuous initial data, Numer. Math., 126(4):703–740, 2014.Google Scholar
[40] Zhu, H. Q. and Qiu, J. X., Adaptive Runge–Kutta discontinuous Galerkin methods using different indicators: one-dimensional case, J. Comput. Phys., 228(18):6957–6976, 2009.Google Scholar