A special class of continuous general linear methods

Abstract

We consider the construction of a class of numerical methods based on the general matrix inverse [14] which provides continuous interpolant for dense approximations (output). Their stability properties are similar to those for Runge-Kutta methods. These methods provide a unifying scope for many families of traditional methods. They are self-starting, to change stepsize during integration is not difficult when using them. We exploited these properties by first obtaining the direct block methods associated with the continuous schemes and then converting the block methods into uniformly A-stable high order general linear methods that are acceptable for solving stiff initial value problems. However, we will limit our formulation only for the step numbers k = 2, 3, 4. From our preliminary experiments we present some numerical results of some initial value problems in ordinary differential equations illustrating various features of the new class of methods. Mathematical subject classification: 65L05.