Efficient solution techniques for implicit finite element schemes with flux limiters

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.