Iterative solutions of mildly nonlinear systems

The correct numerical modelling of free-surface hydrodynamics often requires the solution of diagonally nonlinear systems. In doing this, one may substantially enhance the model accuracy while fulfilling relevant physical constraints. This is the case when a suitable semi-implicit discretization is used, e.g., to solve the one-dimensional or the multi-dimensional shallow water equations; to model axially symmetric flows in compliant arterial systems; to solve the Boussinesq equation in confined–unconfined aquifers; or to solve the mixed form of the Richards equation. In this paper two nested iterative methods for solving a mildly nonlinear system   of the form V(η)+Tη=b are proposed and analysed. It is shown that the inner and the outer iterates are monotone, and converge to the exact solution for a wide class of mildly nonlinear systems of applied interest. A simple, and yet non-trivial test problem derived from the mathematical modelling of flows in porous media is formulated and solved with the proposed methods.