The spatial discretization of nonlinear partial differential equations (PDEs) results in large systems of nonlinear ordinary differential equations (ODEs). The discretization of the Brusselator equation is a characteristic example. For the parallel numerical solution of the Brusselator equation we use an iterated Runge-Kutta method. We propose modifications of the original method that exploit t...

In this paper we discuss the use of stabilized Runge-Kutta methods to accelerate the solution of systems of nonlinear equations. The general idea is to seek solutions as steady state solutions of an associated system of ordinary differential equations. A class of stabilized RungeKutta methods are derived that can be used to efficiently evolve the associated system to steady state. Computational...

This paper presents a proof that the use of polynomial Lyapunov functions is not conservative for studying exponential stability properties of nonlinear ordinary differential equations on bounded regions. The main result implies that if there exists an n-times continuously differentiable Lyapunov function which proves exponential decay on a bounded subset of R, then there exists a polynomial Ly...