A Class of Single Step Methods with a Large Interval of Absolute Stability (Journal of Computational Mathematics, 1991)
In this paper, a class of integration formulas is derived from the approximation so that the first derivative can be expressed within an interval $[nh,(n+1)h]$ as $$\frac{dy}{dt}=-P(y-y_n)+f_n+Q_n(t).$$ The class of formulas is exact if the differential equation has the shown form, where $P$ is a diagonal matrix, whose elements $$-pj=\frac{\partial}{\partial y_j}f_j(t_n,y_n),j=1,…,m$$ are constant in the interval $[nh,(n+1)h]$, and $Q_n(t)$ is a polynomial in $t$. Each of the formulas derived in this paper includes only the first derivative $f$ and $$-pj=\frac{\partial}{\partial y_j}f_j(t_n,y_n).$$ It is identical with a certain Runge-Kutta method. In particular, when $Q_n(t)$ is a polynomial of degree two, one of our formulas is an extension of Treanor's method, and possesses better stability properties. Therefore the formulas derived in this paper can be regarded as a modified or an extended form of the classical Runge-Kutta methods. Preliminary numerical results indicate that our fourth order formula is superior to Treanor's in stability properties.