Topics in matrix analysis
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Iterative Solution of Piecewise Linear Systems
SIAM Journal on Scientific Computing
Iterative Solution of Piecewise Linear Systems and Applications to Flows in Porous Media
SIAM Journal on Scientific Computing
A Nested Newton-Type Algorithm for Finite Volume Methods Solving Richards' Equation in Mixed Form
SIAM Journal on Scientific Computing
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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(@h)+T@h=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.