Some General Linear Methods for the Numerical Solution of Non-Stiff IVPs in ODEs

Abstract

In this paper, we consider the construction of explicit General Linear Methods (GLM) for the numerical solution of non-stiff initial value problems (IVPs) in ordinary differential equations (ODEs). The discrete coefficients of the methods are obtained from the continuous coefficients of the continuous schemes which are derived using collocation and interpolation approach. One of the new method's stability matrix M ( z ) has a single nonzero eigenvalue similar to that of the third order Runge Kutta method (RKM), while other eigenvalues being zero, such general linear method is referred to as GLM with Runge Kutta stability. Numerical results are provided to illustrate the application of the new schemes. The new schemes are found to have competitive accuracy with the existing Almost Runge Kutta Methods (ARKM) discussed in Butcher [9].