A nonlinearity lagging for the solution of nonlinear steady state reaction diffusion problems

This paper concerns with the analysis of the iterative procedure for the solution of a nonlinear reaction diffusion equation at the steady state in a two dimensional bounded domain supplemented by suitable boundary conditions. This procedure, called Lagged Diffusivity Functional Iteration (LDFI)-procedure, computes the solution by "lagging'' the diffusion term. A model problem is considered and a finite difference discretization for that model problem is described.Furthermore, properties of the finite difference operator are proved. Then, sufficient conditions for the convergence of the LDFI-procedure are given. At each stage of the LDFI-procedure a weakly nonlinearalgebraic system has to be solved and the simplified Newton-Arithmetic Mean method is used. This method is particularly well suited for implementation on parallel computers.Numerical studies show the efficiency, for different test functions, of the LDFI-procedure combined with the simplified Newton-Arithmetic Mean method. Better results are obtained when in the reaction diffusion equation also a convection term is present

Lagged diffusivity fixed point iteration for solving steady-state reaction diffusion problems

The paper concerns with the computational algorithms for a steady-state reaction diffusion problem. A lagged diffusivity iterative algorithm is proposed for solving resulting system of quasilinear equations from a finite difference discretization. The convergence of the algorithm is discussed and the numerical results show the efficiency of this algorithm

On the Lagged Diffusivity Method for the solution of nonlinear finite difference systems

In this paper, we extend the analysis of the Lagged Diffusivity Method for nonlinear, non-steady reaction-convection-diffusion equations. In particular, we describe how the method can be used to solve the systems arising from different discretization schemes, recalling some results on the convergence of the method itself. Moreover, we also analyze the behavior of the method in case of problems presenting boundary layers or blow-up solutions

Numerical studies on semi-implicit and implicit methods for reaction-diffusion equations

In this report Rosenbrock, extended and generalized trapezoidal formulae are considered. Numerical studies on these methods have been developed on a linear and a nonlinear reaction diffusion convection equation

the arithmetic mean solver in lagged diffusivity method for nonlinear diffusion equations

Th is paper deals with the solution of nonlinear system arising fro m finite difference discretization of nonlinear diffusion convection equations by the lagged diffusivity functional iteration method co mbined with d ifferent inner iterative solvers. The analysis of the whole procedure with the splitt ing methods of the Arith met ic Mean (AM) and of the Alternating Group Exp licit (A GE) has been developed. A comparison in terms of number of iterations has been done with the BiCG-STA B algorith m. So me nu merical experiments have been carried out and they seem to show the effectiveness of the lagged diffusivity procedure with the Arithmet ic Mean method as inner solver

Hestenes method for symmetric indefinite systems in interior-point method

This paper deals with the analysis and the solution of the Karush-Kuhn-Tucker (KKT) system that arises at each iteration of an Interior-Point (IP) method for minimizing a nonlinear function subject to equality and inequality constraints.This system is generally large and sparse and it can be reduced so that the coefficient matrix is still sparse, symmetric and indefinite, with size equal to the number of the primal variables and of the equality constraints. Instead of transforming this reduced system to a quasidefinite form by regularization techniques used in available codes on IP methods, under standard assumptions on the nonlinear problem, the system can be viewed as the optimality Lagrange conditions for a linear equality constrained quadratic programming problem, so that Hestenes multipliers' method can be applied. Numerical experiments on elliptic control problems with boundary and distributed control show the effectiveness of Hestenes scheme as inner solver for IP methods

The Newton-arithmetic mean method for the solution of systems of nonlinear equations

This paper is concerned with the development of the Newton-arithmetic mean method for large systems of nonlinear equations with block-partitioned Jacobian matrix. This method is well suited for implementation on a parallel computer; its degree of decomposition is very high. The convergence of the method is analysed for the class of systems whose Jacobian matrix satisfies an affine invariant Lipschitz condition. An estimation of the radius of the attraction ball is given. Special attention is reserved to the case of weakly nonlinear systems. A numerical example highlights some peculiar properties of the method. (C) 2002 Elsevier Science Inc. All rights reserved

Iterative solution of elliptic control problems with control and state constraints

This paper concerns with the solution of optimal control problems by means of nonlinear programming methods. The direct transcription,by finite difference approximation, of the optimal control problem into a finite dimensional nonlinear programming problem is described.An iterative procedure for the solution of this nonlinear program is presented. An extensive numerical analysis of the behaviour of the method isreported on boundary control and distributed control problems with boundary conditions of Dirichlet or Neumann or mixed type