Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-16T01:20:30.231Z Has data issue: false hasContentIssue false
  • English
  • Français

Extension Of The High-Order Space-Time Discontinuous Galerkin Cell Vertex Scheme To Solve Time Dependent Diffusion Equations

Published online by Cambridge University Press:  20 August 2015

Shuangzhang Tu ,
Gordon W. Skelton  and
Qing Pang
Shuangzhang Tu*
Affiliation:
Department of Computer Engineering, Jackson State University, Jackson, MS 39217, USA
Gordon W. Skelton*
Affiliation:
Department of Computer Engineering, Jackson State University, Jackson, MS 39217, USA
Qing Pang*
Affiliation:
Department of Computer Engineering, Jackson State University, Jackson, MS 39217, USA
*
Corresponding author.Email:[email protected]
Email:[email protected]
Email:[email protected]
Get access

Abstract

In this paper, the high-order space-time discontinuous Galerkin cell vertex scheme (DG-CVS) developed by the authors for hyperbolic conservation laws is extended for time dependent diffusion equations. In the extension, the treatment of the diffusive flux is exactly the same as that for the advective flux. Thanks to the Riemann-solver-free and reconstruction-free features of DG-CVS, both the advective flux and the diffusive flux are evaluated using continuous information across the cell interface. As a result, the resulting formulation with diffusive fluxes present is still consistent and does not need any extra ad hoc techniques to cure the common “variational crime” problem when traditional DG methods are applied to diffusion problems. For this reason, DG-CVS is conceptually simpler than other existing DG-typed methods.

Keywords

65M9976M25High-order methodspace-time methoddiscontinuous Galerkin (DG) methodcell-vertex scheme (CVS)diffusion equations
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 5 , May 2012 , pp. 1503 - 1524
Copyright
Copyright © Global Science Press Limited 2012

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]Tu, S., A high order space-time Riemann-solver-free method for solving compressible Euler equations, January 2009, AIAA Paper 2009–1335.Google Scholar
[2]Tu, S., A solution limiting procedure for an arbitrarily high order space-time method, June 2009, AIAA Paper 2009–3983.Google Scholar
[3]Tu, S., Skelton, G., and Pang, Q., A compact high order space-time method for conservation laws, Communications in Computational Physics, Vol. 9, No. 2, 2011, pp. 441–480.Google Scholar
[4]Tu, S. and Tian, Z., Preliminary Implementation of a High Order Space-time Method on Overset Cartesian/Quadrilateral Grids, January 2010, AIAA Paper 2010–0544.Google Scholar
[5]Chang, S.-C. and To, W., A new numerical framework for solving conservation laws: the method of space-time conservation element and solution element, 1991, NASA TM 1991–104495.Google Scholar
[6]Baumann, C. E. and Oden, J. T., A discontinuous hp finite element method for convection-diffusion problems, Comput. Methods Appl. Mech. Engrg., Vol. 175, 1999, pp. 311–341.Google Scholar
[7]Cockburn, B. and Shu, C.-W., The local Discontinuous Galerkin Method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., Vol. 35, 1998, pp. 2440–2463.CrossRefGoogle Scholar
[8]Bassi, F. and Rebay, S., A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys., Vol. 131, No. 2, 1997, pp. 267–279.CrossRefGoogle Scholar
[9]Arnold, D. N., Brezzi, F., Cockburn, B., and Marini, D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., Vol. 39, No. 5, 2001, pp. 1749–1779.CrossRefGoogle Scholar
[10]Sudirham, J. J., van der Vegt, J. J. W., and van Dam R., M. J., Space-time discontinuous Galerkin method for advection-diffusion problems on time-dependent domains, Appl. Numer. Math., Vol. 56, December 2006, pp. 1491–1518.CrossRefGoogle Scholar
[11]Klaij, C. M., van der Vegt, J. J. W., and van der Ven, H., Space-time discontinuous Galerkin method for the compressible Navier-Stokes equations, J. Comput. Phys., Vol. 217, September 2006, pp. 589–611.CrossRefGoogle Scholar
[12]Peraire, J. and Persson, P.-O., The compact discontinuous Galerkin (CDG) method for elliptic problems, SIAM J. Sci. Comput., Vol. 30, No. 4, 2008, pp. 1806–1824.CrossRefGoogle Scholar
[13] van Leer, B. and Lo, M., A Discontinuous Galerkin Method for Diffusion Based on Recovery, June 2007, AIAA Paper 2007–4083.Google Scholar
[14]Liu, Y., Shu, C.-W., Tadmor, E., and Zhang, M., Central Local Discontinuous Galerkin Methods on Overlapping Cells for Diffusion Equations, 2010, submitted to Mathematical Modelling and Numerical Analysis and under revision.Google Scholar
[15]Baumann, C. and Oden, J., A discontinuous hp finite element method for convection-diffusion problems, Comput. Methods Appl. Mech. Engrg., Vol. 175, 1999, pp. 311–341.Google Scholar
[16]Babusška, I., Baumann, C., and Oden, J., A discontinuous hp finite element method for diffusion problems: 1-D analysis, Computers and Mathematics with Applications, Vol. 37, No. 9, 1999, pp. 103–122.Google Scholar
[17]Larson, M. G. and Niklasson, A. J., Analysis of a family of discontinuous Galerkin methods for elliptic problems: the one dimensional case, Numer. Math., Vol. 99, November 2004, pp. 113–130.CrossRefGoogle Scholar
[18]Romkes, A., Prudhomme, S., and Oden, J. T., Convergence analysis of a discontinuous finite element formulation based on second order derivatives, Comput. Meth. Appl. Mech. Eng., Vol. 195, 2006, pp. 3461–3482.Google Scholar
[19]Dolejsší, V. and Havle, O., The L2-Optimality of the IIPG Method for Odd Degrees of Polynomial Approximation in 1D, J. Sci. Comput., Vol. 42, 2010, pp. 122–143.Google Scholar
[20]Eriksson, K., Estep, D., Hansbo, P., and Johnson, C., Computational Differential Equations, Cambridge University Press, 1996.Google Scholar
[21]Popescu, M., Shyy, W., and Garbey, M., Finite volume treatment of dispersion-relation-preserving and optimized prefactored compact schemes for wave propagation, J. Comput. Phys., Vol. 210, No. 2, 2005, pp. 705–729.Google Scholar