Numer. Math. (2008) 110:405–447 DOI 10.1007/s00211-008-0178-2 Numerische Mathematik Analysis of a BDF–DGFE scheme for nonlinear convection–diffusion problems Vít Dolejší · Miloslav Vlasák Received: 20 June 2007 / Revised: 22 July 2008 / Published online: 16 September 2008 © Springer-Verlag 2008 Abstract We deal with the numerical solution of a scalar nonstationary nonlin- ear convection–diffusion equation. We employ a combination of the discontinuous Galerkin finite element method for the space semi-discretization and the k -step back- ward difference formula for the time discretization. The diffusive and stabilization terms are treated implicitly whereas the nonlinear convective term is treated by a higher order explicit extrapolation method, which leads to the necessity to solve only a linear algebraic problem at each time step. We analyse this scheme and derive a priori asymptotic error estimates in the discrete L ∞ ( L 2 )-norm and the L 2 ( H 1 )-seminorm with respect to the mesh size h and time step τ for k = 2, 3. Numerical examples verifying the theoretical results are presented. Mathematics Subject Classification (2000) 65M15 · 65M12 · 65M60 · 65L06 1 Introduction Our aim is to develop a sufficiently efficient, robust and accurate numerical scheme for the simulation of unsteady viscous compressible flows. During the last years, the This work is a part of the research project MSM 0021620839 financed by the Ministry of Education of the Czech Republic and was partly supported by the Grant No. 316/2006/B-MAT/MFF of the Grant Agency of the Charles University Prague. The research of M. Vlasák was supported by the project LC06052 of the Ministry of Education of the Czech Republic (Jindˇ rich Neˇ cas Center for Mathematical Modelling). V. Dolejší (B ) · M. Vlasák Faculty of Mathematics and Physics, Charles University Prague, Sokolovská 83, 18675 Praha 8, Czech Republic e-mail: dolejsi@karlin.mff.cuni.cz M. Vlasák e-mail: vlasakmila@gmail.com 123