Abstract
We deal with the numerical solution of a scalar nonstationary nonlinear convection-diffusion equation. We present a scheme which uses a discontinuous Galerkin finite element method for a space semi-discretization and the resulting system of ordinary differential equations is discretized by backward difference formulae. The linear terms are treated implicitly whereas the nonlinear ones by a higher order explicit extrapolation which preserves the accuracy of the schemes and leads to a system of linear algebraic equations at each time step. Thenumerical examples presented verify expected orders of convergence.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2002)
Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys. 131, 267–279 (1997)
Baumann, C.E., Oden, J.T.: A discontinuous hp finite element method for the Euler and Navier-Stokes equations. Int. J. Numer. Methods Fluids 31(1), 79–95 (1999)
Cockburn, B.: Discontinuous Galerkin methods for convection dominated problems. In: T.J. Barth, H. Deconinck (eds.) High-Order Methods for Computational Physics, Lecture Notes in Computational Science and Engineering 9, pp. 69–224. Springer, Berlin (1999)
Dolejšý, V.: On the discontinuous Galerkin method for the numerical solution of the Navier-Stokes equations. Int. J. Numer. Methods Fluids 45, 1083–1106 (2004)
Dolejšý, V., Feistauer, M.: Error estimates of the discontinuous Galerkin method for nonlinear nonstationary convection-diffusion problems. Numer. Funct. Anal. Optim. 26(25-26), 2709–2733 (2005)
Dolejšý, V., Feistauer, M., Hozman, J.: Analysis of semi-implicit DGFEM for nonlinear convection-diffusion problems. Comput. Methods Appl. Mech. Engrg. (submitted). Preprint No. MATH-knm-2005/4, Charles University Prague, School of Mathematics, 2005
Dolejšý, V., Feistauer, M., Sobotýkováá, V.: A discontinuous Galerkin method for nonlinear convection-diffusion problems. Comput. Methods Appl. Mech. Engrg. 194, 2709–2733 (2005)
Feistauer, M., Felcman, J., Straškraba, I.: Mathematical and Computational Methods for Compressible Flow. Oxford University Press, Oxford (2003)
Lomtev, I., Quillen, C.B., Karniadakis, G.E.: Spectral/hp methods for viscous compressible flows on unstructured 2d meshes. J. Comput. Phys 144(2), 325–357 (1998)
van der Vegt, J.J.W., 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(2), 546–585 (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer
About this paper
Cite this paper
Dolejší, V. (2006). Higher Order Semi-Implicit Discontinuous Galerkin Finite Element Schemes for Nonlinear Convection-Diffusion Problems. In: de Castro, A.B., Gómez, D., Quintela, P., Salgado, P. (eds) Numerical Mathematics and Advanced Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-34288-5_38
Download citation
DOI: https://doi.org/10.1007/978-3-540-34288-5_38
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-34287-8
Online ISBN: 978-3-540-34288-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)