Volume 9, Issue 2
PA Class of Single Step Methods with a Large Interval of Absolute Stability

Geng Sun

DOI:

J. Comp. Math., 9 (1991), pp. 185-193

Published online: 1991-09

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  • Abstract

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 $-pj=\frac{\partial}{\partial y_j}f_j(t_n,y_n),j=1,\cdots,m$ 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.

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@Article{JCM-9-185, author = {}, title = {PA Class of Single Step Methods with a Large Interval of Absolute Stability}, journal = {Journal of Computational Mathematics}, year = {1991}, volume = {9}, number = {2}, pages = {185--193}, abstract = { 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 $-pj=\frac{\partial}{\partial y_j}f_j(t_n,y_n),j=1,\cdots,m$ 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. }, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9391.html} }
TY - JOUR T1 - PA Class of Single Step Methods with a Large Interval of Absolute Stability JO - Journal of Computational Mathematics VL - 2 SP - 185 EP - 193 PY - 1991 DA - 1991/09 SN - 9 DO - http://dor.org/ UR - https://global-sci.org/intro/jcm/9391.html KW - AB - 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 $-pj=\frac{\partial}{\partial y_j}f_j(t_n,y_n),j=1,\cdots,m$ 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.
Geng Sun. (1970). PA Class of Single Step Methods with a Large Interval of Absolute Stability. Journal of Computational Mathematics. 9 (2). 185-193. doi:
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