Two-step almost collocation methods for Volterra integral equations

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

“…However, for small values of ( = 10 −6 ) for which the van der Pol oscillator is stiff the Runge-Kutta-Gauss Table 5 Numerical results for Runge-Kutta-Gauss method of order p = 4 and stage order q = 2 for the Hires problem 6 .42 · 10 −7 2.25 10 1.47 · 10 −7 2.12 11 3.52 · 10 −8 2.06 12 8.62 · 10 −9 2.03 method exhibits order reduction phenomenon and its order of convergence drops to about p = 2 which corresponds to the stage order q = 2. This is not the case for TSRK method which preserves order of convergence p = q = 4, which leads to higher accuracy.…”

“…Different approaches to the construction of continuous TSRK methods are presented in [4], Bartoszewski et al (unpublished manuscript) and [19]. TSRK methods for delay differential equations are considered in [2,5] and for Volterra integral equations in [10].…”

Two-step almost collocation methods for ordinary differential equations

“…Different approach to the construction of continuous two-step RungeKutta methods is presented in [4,15] and Bartoszewski et al (unpublished manuscript). Continuous two-step Runge-Kutta methods for delay differential equations are considered in [2,5] and for Volterra integral equations in [10].…”

A Family of Multistep Collocation Methods for Volterra Integral Equations

“…In [7] we introduce a modification in the technique also fo Volterra integro-differential equations, by relaxing some of the collocation conditions and by introducing some previous stage values, in order to further increase the order and to have free parameters in the method, to be used to get A-stability, considering also implementation issues. We didn't find A-stable methods within this class, but wide stability regions exist.…”