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A family of \(L\)-stable singly implicit peer methods for solving stiff IVPs. (English) Zbl 07074123
Summary: 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.
MSC:
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L04 Numerical methods for stiff equations
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