INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2007; 55:611–635 Published online 20 March 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/fld.1470 Efficient solution techniques for implicit finite element schemes with flux limiters M. M¨ oller ∗, † Institute of Applied Mathematics (LS III), University of Dortmund, Vogelpothsweg 87, Dortmund D-44227, Germany SUMMARY The algebraic flux correction (AFC) paradigm is equipped with efficient solution strategies for implicit time-stepping schemes. It is shown that Newton-like techniques can be applied to the nonlinear systems of equations resulting from the application of high-resolution flux limiting schemes. To this end, the Jacobian matrix is approximated by means of first- or second-order finite differences. The edge-based formulation of AFC schemes can be exploited to devise an efficient assembly procedure for the Jacobian. Each matrix entry is constructed from a differential and an average contribution edge by edge. The perturbation of solution values affects the nodal correction factors at neighbouring vertices so that the stencil for each individual node needs to be extended. Two alternative strategies for constructing the corresponding sparsity pattern of the resulting Jacobian are proposed. For nonlinear governing equations, the contribution to the Newton matrix which is associated with the discrete transport operator is approximated by means of divided differences and assembled edge by edge. Numerical examples for both linear and nonlinear benchmark problems are presented to illustrate the superiority of Newton methods as compared to the standard defect correction approach. Copyright 2007 John Wiley & Sons, Ltd. Received 29 August 2006; Revised 31 January 2007; Accepted 1 February 2007 KEY WORDS: Newton-like solution techniques; sparse Jacobian evaluation; high-resolution schemes; flux-corrected transport; finite elements 1. INTRODUCTION For decades, the development of reliable discretization techniques for convection-dominated flows has been one of the primary interests in computational fluid dynamics. A variety of stabilization techniques and high-resolution schemes based on flux/slope limiting have been presented in the literature to combat the formation of non-physical oscillations which would be generated otherwise. ∗ Correspondence to: M. M¨ oller, Institute of Applied Mathematics (LS III), University of Dortmund, Vogelpothsweg 87, Dortmund D-44227, Germany. † E-mail: matthias.moeller@math.uni-dortmund.de Copyright 2007 John Wiley & Sons, Ltd.