Nordsieck representation of two-step Runge–Kutta methods for ordinary differential equations

Continuous two-step Runge–Kutta methods for ordinary differential equations

“…TSRK methods were introduced by Jackiewicz and Tracogna [18] and further investigated in [1,3,8,9,15,17,20,25], Conte et al (unpublished manuscript) and [26]. Continuous methods (1.1) provide an approximation to the solution y(t) of (1.2) on the whole interval of integration, and not only in the gridpoints {t n } as in the case of discrete TSRK methods (1.3).…”

Two-step almost collocation methods for ordinary differential equations

“…This general class of TSRK methods was introduced by Jackiewicz and Tracogna [14] and further investigated in [1,3,8,9,12,16,26], Conte et al (unpublished manuscript), and [27]. The special case of collocation methods (1.2) provide continuous approximation to the solution y(t) of (1.1) on the whole interval of integration, and not only at the gridpoints {t n } as is the case for the methods defined by (1.3).…”

Order conditions for general linear methods

“…, N − 1, and the vectors q 0 and q 1 are [3,9]). TSRK methods (2.6) were introduced by Jackiewicz and Tracogna [39], and further investigated in [35,28,[36][37][38][40][41][42][43][44][45] and the monograph [3].…”

“…GLMs were first introduced by Butcher [4] using somewhat different notation, and the modern theory of these methods was developed in [5][6][7]2,8,3,9]. These methods include as special cases many numerical methods for ODEs, for example, linear multistep methods [2,8,[10][11][12][13][14], predictor-corrector methods [11][12][13][14], Runge-Kutta (RK) methods [5,6,15,2,11,12], diagonally implicit multistage integration methods (DIMSIMs) [16][17][18][19][20][21][22][23][24][25][26][27], Nordsieck methods [28][29][30][31][32][33][34], two-step Runge-Kutta (TSRK) methods [35,28,[36][37][38]…”