Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-28T15:48:58.940Z Has data issue: false hasContentIssue false
  • English
  • Français

A Comparison of Semi-Lagrangian and Lagrange-Galerkin hp-FEM Methods in Convection-Diffusion Problems

Published online by Cambridge University Press:  20 August 2015

Pedro Galán del Sastre  and
Rodolfo Bermejo
Pedro Galán del Sastre*
Affiliation:
Departamento de Matemática Aplicada al Urbanismo, a la Edificación y al Medio Ambiente, E.T.S.A.M., Universidad Politécnica de Madrid, Avda. Juan de Herrera 4, 28040 Madrid, Spain
Rodolfo Bermejo*
Affiliation:
Departamento de Matematica Aplicada, E.T.S.I.I., Universidad Politécnica de Madrid, C/ José Gutiérrez Abascal 2, 28006 Madrid, Spain
*
Corresponding author.Email:[email protected]
Email:[email protected]
Get access

Abstract

We perform a comparison in terms of accuracy and CPU time between second order BDF semi-Lagrangian and Lagrange-Galerkin schemes in combination with high order finite element method. The numerical results show that for polynomials of degree 2 semi-Lagrangian schemes are faster than Lagrange-Galerkin schemes for the same number of degrees of freedom, however, for the same level of accuracy both methods are about the same in terms of CPU time. For polynomials of degree larger than 2, Lagrange-Galerkin schemes behave better than semi-Lagrangian schemes in terms of both accuracy and CPU time; specially, for polynomials of degree 8 or larger. Also, we have performed tests on the parallelization of these schemes and the speedup obtained is quasi-optimal even with more than 100 processors.

Keywords

65M6065L6065N3065M2576M1076M25Navier-Stokes equationsconvection-diffusion equationssemi-LagrangianLagrange-Galerkinsecond order backward difference formulahp-finite element method
Type
Research Article
Information
Communications in Computational Physics , Volume 9 , Issue 4 , April 2011 , pp. 1020 - 1039
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]Bermejo, R. and Carpio, J., A semi-Lagrangian-Galerkin projection scheme for convection equations, IMA J. Numer. Anal., 30(3) (2010), 799–831.Google Scholar
[2]Boukir, K., Maday, Y., Meétivet, B., and Razafindrakoto, E., A high-order characteristics/finite element method for the incompressible Navier-Stokes equations, Int. J. Numer. Methods. Fluids., 25(12) (1997), 1421–1454.3.0.CO;2-A>CrossRefGoogle Scholar
[3]Bramble, J. H., Pasciak, J. E., and Vassilev, A. T., Uzawa type algorithms for nonsymmetric saddle point problems, Math. Comput., 69(230) (2000), 667–689.Google Scholar
[4]Dean, E. J. and Glowinski, R., On some finite element methods for the numerical simulation of incompressible viscous flow, in Incompressible Computational Fluid Dynamics, pages 17–65, Cambridge University Press, 1993.Google Scholar
[5]Douglas, J. Jr. and Russell, T. F., Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal., 19(5) (1982), 871–885.Google Scholar
[6]Erturk, E., Corke, T. C., and Gokcol, C., Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers, Int. J. Numer. Meth. Fluids., 48(7) (2005), 747–774.Google Scholar
[7]Ewing, R. E. and Russell, T. F., Multistep Galerkin methods along characteristics for convection-diffusion problems, Advances in Computer Methods for Partial Differential Equations-IV, pages 28–36, 1981.Google Scholar
[8]Ghia, U., Ghia, N., and Shin, C. T., High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, J. Comput. Phys., 48 (1982), 387–411.Google Scholar
[9]Giraldo, F. X., The Lagrange-Galerkin spectral element method on unstructured quadrilateral grids, J. Comput. Phys., 147(1) (1998), 114–146.Google Scholar
[10]Guermond, J. L., Minev, P., and Shen, J., Anoverviewof projection methods for incompressible flows, Comput. Methods. Appl. Mech. Engrg., 195(44-47) (2006), 6011–6045.CrossRefGoogle Scholar
[11]Guermond, J. L. and Shen, J., A new class of truly consistent splitting schemes for incompressible flows, J. Comput. Phys., 192(1) (2003), 262–276.Google Scholar
[12]Guermond, J. L. and Shen, J., Velocity-correction projection methods for incompressible flows, SIAM J. Numer. Anal., 41(1) (2003), 112–134 (electronic).Google Scholar
[13]Karniadakis, G. E., Israeli, M., and Orszag, S. A., High-order splitting methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 97(2) (1991), 414–443.Google Scholar
[14]Karniadakis, G. E. and Sherwin, S. J., Spectral/hp Element Methods for CFD,Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 1999.Google Scholar
[15]Morton, K. W., Priestley, A., and Süli, E., Stability of the Lagrange-Galerkin method with nonexact integration, RAIRO Modél. Math. Anal. Numér., 22(4) (1988), 625–653.Google Scholar
[16]Pironneau, O., On the transport-diffusion algorithm and its applications to the Navier-Stokes equations, Numer. Math., 38(3) (1981/82), 309–332.CrossRefGoogle Scholar
[17]Temperton, C. and Staniforth, A., An efficient two-time-level semi-lagrangian semi-implicit integration scheme, Q.J.R. Meteorol. Soc., 113(477) (1987), 1025–1039.Google Scholar
[18]Šolín, P., Segeth, K., and Dolezžel, I., Higher-Order Finite Element Methods,Studies in Advanced Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2004.Google Scholar
[19]Xiu, D. and Karniadakis, G. E., A semi-Lagrangian high-order method for Navier-Stokes equations, J. Comput. Phys., 172(2) (2001), 658–684.Google Scholar