The {/it 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 algebraic flux correction 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 neighboring 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.