Diagonally implicit general linear methods for ordinary differential equations (Q1314633)

The authors construct linear approximation methods of the type: \((*)\) \(Y_ i= h\sum^ s_{j=1} a_{ij} f(Y_ j)+ \sum u_{ij} y_ j^{[n-1]}\), \(j=1,2,\dots,s\), \(y_ i^{[n]}= h\sum^ s_{j=1} b_{ij} f(Y_ j)+ \sum v_{ij} y_ j^{[n-1]}\), \(i=1,2,\dots,r\), for \(y'(x)= f(y(x))\), \(x\in [x_ 0,X]\), \(y(x_ 0)= y_ 0\). A method of the type \((*)\) is said to have \(s\) internal stages and \(r\) external approximations. This class \((*)\) includes many known numerical methods such as Runge-Kutta, multistep, etc.. Suppose \(q\) is the stage approximation order and \(p\) that of the external approximations. The authors construct new interesting methods corresponding to the cases: \(s+ 1= r=q\), \(p=q\) or \(p= q+1\); \(s= r+1= q\), \(p= q\) or \(p= q+1\) and \(s= r= q\), \(p= q+1\). Effective methods are given and their stability properties investigated.