Linear Multistep Methods—I: Construction and Consistency

The effectiveness of the family of TS(p) methods has been evident in the preceding chapter. For order p > 1, however, they suffer a serious disadvantage in that they require the right-hand side of the differential equation to be differentiated a number of times. This often rules out their use in real-world applications, which generally involve (large) systems of ODEs whose differentiation is impractical unless automated tools are used [23]. We look, therefore, for alternatives that do not require the use of second and higher derivatives of the solution.