Modified Nyström Methods for Semi-discrete Hyperbolic Differential Equations

Abstract

First and second order Nyström type methods are derived for second order differential equations, without first derivatives, possessing the following properties: (i) the stability interval equals $4m^2 ,m$ denoting the number of stages per integration step; (ii) the method is internally stable irrespective of the value of m; (iii) the storage requirements are limited; and (iv) the costs per integration step are one right-hand side evaluation, one evaluation of the Jacobian matrix and $m - 1$ matrix-vector multiplications. These four properties are of interest in the integration of the usually very large systems of ordinary differential equations resulting from the semi-discretization of partial differential equations which are of hyperbolic type and of second order in time.