Jacobian-Dependent Two-Stage Peer Method for Ordinary Differential Equations

Two-step peer methods with equation-dependent coefficients

“…Moreover, this paper also completes the work done in Conte et al (2021), where the coefficients of the proposed methods have been derived only in the scalar case and when the accuracy order and the number of stages are equal to two. Now, we derive the old methods coefficients (which become matrices) also for non-scalar problems.…”

“…To understand why the error associated with Y i takes this form, just extend the procedure shown in Conte et al (2021) for err 1 (8), observing that it depends on the moment L 1,3 . Still referring to the simple two-stage case, look at Eq.…”

“…Specifically, this paper is organized as follows: in Sect. 2, we extend the work done in Conte et al (2021), formulating the OEJDP methods in the non-scalar case and deriving the related order conditions for p = s ≥ 2; in Sect. 3, we derive two-stage NEJDP methods of order two, showing the relative linear stability properties; in Sect.…”

“…We were inspired by Ixaru (2012), in which a similar procedure was applied to explicit Runge-Kutta methods improving their accuracy and stability properties, as evidenced also by the numerous numerical tests conducted in Conte et al (2020a). An extension of this methodology has also been applied to peer methods in Conte et al (2021), in which the authors have managed to derive Jacobiandependent coefficients, slightly improving the stability properties of classical schemes. In this work, we show that by changing the approach proposed in Conte et al (2021), it is possible to obtain New Explicit Jacobian-Dependent Peer (NEJDP) methods with much better stability and accuracy properties than those obtained in Conte et al (2021), which will be called Old Explicit Jacobian-Dependent Peer (OEJDP) methods.…”

“…Therefore, Sect. 2 of this paper is devoted to the extension of OEJDP methods of Conte et al (2021), while the following ones are devoted to the derivation of NEJDP methods.…”