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 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.