Hostname: page-component-7bb8b95d7b-s9k8s Total loading time: 0 Render date: 2024-09-18T17:51:16.419Z Has data issue: false hasContentIssue false
  • English
  • Français

A Compact High Order Space-Time Method for Conservation Laws

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

This paper presents a novel high-order space-time method for hyperbolic conservation laws. Two important concepts, the staggered space-time mesh of the space-time conservation element/solution element (CE/SE) method and the local discontinuous basis functions of the space-time discontinuous Galerkin (DG) finite element method, are the two key ingredients of the new scheme. The staggered space-time mesh is constructed using the cell-vertex structure of the underlying spatial mesh. The universal definitions of CEs and SEs are independent of the underlying spatial mesh and thus suitable for arbitrarily unstructured meshes. The solution within each physical time step is updated alternately at the cell level and the vertex level. For this solution updating strategy and the DG ingredient, the new scheme here is termed as the discontinuous Galerkin cell-vertex scheme (DG-CVS). The high order of accuracy is achieved by employing high-order Taylor polynomials as the basis functions inside each SE. The present DG-CVS exhibits many advantageous features such as Riemann-solver-free, high-order accuracy, point-implicitness, compactness, and ease of handling boundary conditions. Several numerical tests including the scalar advection equations and compressible Euler equations will demonstrate the performance of the new method.

Keywords

65M9976M25High order methodspace-time methodcell-vertex scheme (CVS)conservation laws
Type
Research Article
Information
Communications in Computational Physics , Volume 9 , Issue 2 , February 2011 , pp. 441 - 480
Copyright
Copyright © Global Science Press Limited 2011

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]Arbogast, T. and Wheeler, M., A characteristics-mixed finite element method for advection-dominated transport problems, SIAM J. Numer. Anal., 32 (1995), 404–424.CrossRefGoogle Scholar
[2]Chang, S.-C., Courant number and Mach number insensitive CE/SE Euler solvers, AIAA Paper 2005-4355Google Scholar
[3]Chang, S.-C. and To, W., A new numerical framework for solving conservation laws: the method of space-time conservation element and solution element, NASA. TM., 1991-104495.Google Scholar
[4]Chang, S.-C. and Wang, X., Multi-dimensional Courant number insensitive CE/SE Euler solvers for applications involving highly nonuniform meshes, AIAA Paper 2003-5285.CrossRefGoogle Scholar
[5]Chang, S.-C., Wang, X. and Chow, C., The space-time conservation element and solution element method: a new high-resolution and genuinely multidimensional paradigm for solving conservation laws, J. Comput. Phys., 156(1) (1999), 89–136.Google Scholar
[6]Cockburn, B. and 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
[7]Cockburn, B. and Shu, C.-W., Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16(3) (2001), 173–261.CrossRefGoogle Scholar
[8]Dasgupta, G., Integration within polygonal finite elements, J. Aerosp. Engrg., 16(1) (2003), 9–18.CrossRefGoogle Scholar
[9]Dawson, C., Godunov-mixed methods for advection-diffusion equations in multidimen-sions, SIAM J. Numer. Anal., 30 (1993), 1315–1332.Google Scholar
[10]Gressier, J. and Moschetta, J., Robustness versus accuracy in shock-wave computations, Int. J. Numer. Meth. Fluids., 33(1) (2000), 313–332.Google Scholar
[11]Jiang, G. and Shu, C.-W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126 (1996), 202–228.CrossRefGoogle Scholar
[12]Kurganov, A. and Tadmor, E., New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 160(1) (2000), 241–282.Google Scholar
[13]Lesoinne, M. and Farhat, C., Geometric conservation laws for flow problems with moving boundaries and deformable meshes, and their impact on aeroelastic computations, Comput. Meth. Appl. Mech. Eng., 134(1) (1996), 71–90.Google Scholar
[14]Liu, X.-D., Osher, S. and Chan, T., Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115(1) (1994), 200–212.Google Scholar
[15]Liu, Y., Central schemes on overlapping cells, J. Comput. Phys., 209(1) (2005), 82–104.Google Scholar
[16]Loh, C. Y., Hultgren, L. S. and Chang, S.-C., Wave computation in compressible flow using space-time conservation element and solution element method, AIAA J., 39(5) (2001), 794–801.CrossRefGoogle Scholar
[17]Lowrie, R., Roe, P., and van Leer, B., A space-time discontinuous Galerkin method for the time-accurate numerical solution of hyperbolic conservation laws, AIAA Paper 1995-1658.Google Scholar
[18]Nakata, M., The MPACK: Multiple Precision Arithmetic BLAS (MBLAS) and LAPACK (MLAPACK), 2009, http://mplapack.sourceforge.net/.Google Scholar
[19]Nessyahu, H. and Tadmor, E., Non-oscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys., 87 (1990), 408–463.Google Scholar
[20]Palaniappan, J., Haber, R. B. and Jerrard, R. L., A spacetime discontinuous Galerkin method for scalar conservation laws, Comput. Meth. Appl. Mech. Eng., 193(33-35)(2004), 3607–3631.CrossRefGoogle Scholar
[21]Pandolfi, M. and D’Ambrosio, D., Numerical instablilities in upwind methods: analysis and cures for the “carbuncle” phenomenon, J. Comput. Phys., 166(2) (2001), 271–301.Google Scholar
[22]Patera, A., A spectral element method for fluid dynamics-laminar flow in a channel expansion, J. Comput. Phys., 54 (1984), 468–488.Google Scholar
[23]Rathod, H. and Rao, H., Integration of trivariate polynomials over linear polyhedra in Euclidean three-dimensional space, J. Austral. Math. Soc. Ser., 39 (1998), 355–385.CrossRefGoogle Scholar
[24]Robinet, J.-C., Gressier, J., Casalis, G. and Moschetta, J.-M., Shock wave instability and the carbuncle phenomenon: same intrinsic origin, J. Fluid Mech., 417 (2000), 237–263.CrossRefGoogle Scholar
[25]Shu, C.-W., High Order Finite Difference and Finite Volume WENO Schemes and Discontinuous Galerkin Methods for CFD, May 2001, NASA/CR-2001-210865.Google Scholar
[26]Tu, S., A high order space-time Riemann-solver-free method for solving compressible Euler equations, January 2009, AIAA Paper 2009-1335.Google Scholar
[27]Tu, S., A solution limiting procedure for an arbitrarily high order space-time method, June 2009, AIAA Paper 2009-3983.Google Scholar
[28]Tu, S. and Aliabadi, S., A space-time upwind cell-vertex scheme for conservation laws: a Rie-mann solver-free approach, In Bathe, K., editor, Proceedings of the Third M.I.T. Conference on Computational Fluid and Solid Mechanics, pages 1191-1195, Elsevier Ltd., 2005.Google Scholar
[29]Tu, S. and Tian, Z., Preliminary implementation of a high order space-time method on overset cartesian/quadrilateral grids, January 2010, AIAA Paper 2009-0544.Google Scholar
[30]van der Vegt, J. and van der Ven, H., Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows. I. General formulation, J. Comput. Phys., 182 (2002), 546–585.Google Scholar
[31]Wang, Z., Spectral (finite) volume method for conservation laws on unstructured grids, J. Comput. Phys., 178 (2002), 210–251.Google Scholar