Sequence Of Nonlinear Systems Research Articles

An alternating direction Galerkin method for elliptic problems with gradient nonlinearity

AbstractSeveral problems in many applications involve the solution of partial differential equations with gradient dependent nonlinearity. The numerical solution of the resulting nonlinear system is rather expensive. We present an alternating direction Galerkin method that allows much faster solution of the nonlinear system. The alternating direction formulation help reduce the problem into a sequence of nonlinear systems that may be solved very efficiently. Theoretical study of the convergence of the method will also be presented. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Partitioning ODE systems with an application to air pollution models

The numerical treatment of stiff ODE systems is carried out by using implicit methods. A long sequence of nonlinear systems has to be treated when implicit methods are used. The Newton iterative method is often used in the solution of these systems. This leads to the calculation of Jacobian matrices and to the inversion of these matrices. The computational work can in some cases be reduced considerably when some kind of partitioning is used. The conditions under which the partitioning procedures can successfully be used will be studied in this paper. An example, taken from a large air pollution model, will be given to illustrate the usefulness of the theoretical results.

Waveform relaxation methods for implicit differential equations

We apply a Runge-Kutta-based waveform relaxation method to initial-value problems for implicit differential equations. In the implementation of such methods, a sequence of nonlinear systems has to be solved iteratively in each step of the integration process. The size of these systems increases linearly with the number of stages of the underlying Runge-Kutta method, resulting in high linear algebra costs in the iterative process for high-order Runge-Kutta methods. In our earlier investigations of iterative solvers for implicit initial-value problems, we designed an iteration method in which the linear algebra costs are almost independent of the number of stages when implemented on a parallel computer system. In this paper, we use this parallel iteration process in the Runge-Kutta waveform relaxation method. In particular, we analyse the convergence of the method. The theoretical results are illustrated by a few numerical examples.