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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
##### Keywords:
peer two-step methods; multistep methods; L-stability
##### Software:
LSODE; Matlab; MATLAB ODE suite; ode113; Ode15s; ode23; ode23s; ode45; RODAS
Full Text:
##### References:
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