Two-step peer methods with equation-dependent coefficients

Abstract

We introduce a new class of explicit two-step peer methods with the aim of improving the stability properties of already existing peer methods, by making use of coefficients depending on the Jacobian of the Ordinary Differential Equations (ODEs) system to solve. Numerical tests highlight the best stability and accuracy properties of the new methods compared to the classical and equation-dependent ones proposed in Conte et al. (Lect Notes Comput Sci 12949:309–324, 2021).

Explicit two-step peer methods with reused stages

“…We have solved it with the 4-stage peer methods deduced in the previous section, the DOPRI5 and the sixth-order Adams-Bashforth-Moulton (ABM6) to show the behaviour of this class of methods in high dimension problems. with the horizontal force F y (t) = 1/(cosh(4t − 2.5)) 4 and the vertical force F x (t) = 0.4. The initial conditions are θ l (0) = θ(0) = 0, the integration interval is [0, 3.723] and choosing n = 40 we obtain a system of ODEs of dimension 80.…”

“…where ⊗ denotes the standard Kronecker product and I m is the unit matrix of order m. For our studies of order and stability it will be sufficient to consider the scalar case (m = 1) in which (4) becomes (5) In general, two-step s-stage peer methods require s derivative function calls per step. Nevertheless, Horváth and coworkers [11] and Klinge and coworkers [12] have shown that if the matrices A, B and R have a special structure, it is possible to employ less function calls by re-using previously computed stages from the previous steps in the current one, in a similar way as Runge-Kutta schemes do with "first-same-as-last"(FSAL) technique [6].…”

“…, 8 in which three stages are reused and only s − 3 evaluations of the vector field are required. It is worth to remark that after the introduction of peer methods (4) in 2004 by Schmitt and Weiner [13] there has been a constant flow of publications dealing not only with theoretical properties such as accuracy and stability of different families of these type of methods but also with the implementation comparing with classical IVPs solvers for different types of problems. A complete list of publications of peer type methods that includes more than 70 papers can be found in https://www.mathematik.uni-marburg.de/ schmitt/peer/.…”

“…Another recent contribution in the class of ETSPeer methods due to D. Conte and co workers [4], [5], is the development of methods where the coefficients depend on the Jacobian f y (t, y) of the function f (t, y) at suitable points. Here the introduction of this dependence allows the authors to obtain methods with stability properties in all stages similar to linearly implicit methods and therefore are suitable for solving stiff systems.…”

“…Here the introduction of this dependence allows the authors to obtain methods with stability properties in all stages similar to linearly implicit methods and therefore are suitable for solving stiff systems. A detailed derivation of methods and numerical experiments can be seen in [4].…”