Iterative solutions of mildly nonlinear systems

作者：Casulli Vincenzo^*; Zanolli Paola

来源：Journal of Computational and Applied Mathematics, 2012, 236(16): 3937-3947.

DOI：10.1016/j.cam.2012.02.042

• Mildly nonlinear systems
• Functions of bounded variations
• Nested iterations
• Free-surface hydrodynamics
• Wetting and drying
• Confined-unconfined aquifers

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(eta) + T eta = 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.

• 出版日期2012-10