A family of L-stable singly implicit peer methods for solving stiff IVPs

Abstract

In this paper a one parameter family of s-stage singly implicit two-step peer (SIP) methods with order $(s-1)$ that are L-stable for some values of this parameter addressed for the numerical solution of stiff IVPs has been developed. General peer methods are multistage two-step methods for solving IVPs where all stages possess essentially the same accuracy and stability properties. In particular a s-stage SIP requires at each step the solution of s-implicit non-linear systems of equations of the same type in a similar way to the singly implicit Runge–Kutta methods. Here for each $s \ge 3$ a family of one parameter s-stage SIP methods with order $(s-1)$ that are optimally zero-stable for arbitrary step size sequences is derived. For $s \le 8$ intervals of values of this parameter that ensure their L-stability are obtained. Hence for $s \le 8$, L-stable methods with order $(s-1)$ and a computational cost per step equivalent to s one-step backward Euler methods with the same Jacobian matrix are obtained. Further it is shown that under some restriction on the parameter each s-stage SIP method can be formulated as a cyclic multistep method of order $(s-1)$ and this implies that the Dahlquist barrier of second-order for A-stable linear multistep methods can be broken with suitable s-stage cyclic methods of these families.