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A general linear method advances an ordinary differential equation using internal stages $Y$ and a retained external vector $y^{[n]}$. For an autonomous equation $y^{\prime }=f(y)$, its block form isWith $s$ stages and $r$ external components, $A,U,B,V$ have sizes $s\times s$, $s\times r$, $r\times s$, $r\times r$ respectively.

The retained components can encode previous values or derivative information. Runge-Kutta and linear multistep methods fit this framework. With one scalar stage and external component, $A=0$, $U=B=V=1$ gives explicit Euler. The word linear describes these combinations, not a requirement that the differential equation itself be linear.^[ref]

John C. Butcher works on numerical methods for ordinary differential equations, particularly Runge-Kutta methods and general linear methods. His research connects the algebraic structure of a numerical method with its approximation and numerical stability properties. The Butcher tableau records the stages and weights of a Runge-Kutta method compactly.^[ref]

Homepage: www.math.auckland.ac.nz/~butcher/biography.html

Zdzislaw Jackiewicz is a mathematician working in numerical analysis whose work includes methods for ordinary differential equations, delay differential equations and Volterra integral equations. His authored account of general linear methods describes a framework combining internal stage evaluations with a vector of information passed between successive time steps.^[ref]

For $y^{\prime }=f(y)$, a Runge-Kutta method passes an approximation to $y$ between steps while computing several intermediate stages. A linear multistep method instead retains values from several earlier times. General linear methods put both patterns in one matrix formulation, allowing accuracy and numerical stability to be designed together. His Arizona State University profile identifies this subject and the construction of implicit-explicit methods as research interests.^[ref]