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P-stability properties of Runge-Kutta methods for delay differential equations. (English) Zbl 0598.65056

A class of Runge-Kutta methods for delay differential equations (DDE) is studied having in mind the test equation \(y'(t)=ay(t)+by(t-\tau)\), \(t>0\); \(y(t)=g(t)\), \(-\tau \leq t\leq 0\) in case of absolute asymptotic stability (i.e. \(| b| <-Re(a)\) which guarantees that y(t)\(\to 0\), \(t\to \infty\), for all \(\tau >0)\). The methods use certain constrained meshes which allows to get optimal order results. It is shown that any A-stable one-step collocation method for ordinary differential equations without delay inherits the same property being applied to DDE, i.e. it is P- stable. Also, a general approach to find the P-stability regions of Runge-Kutta methods for DDE is presented.
Reviewer: M.M.Konstantinov

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L07 Numerical investigation of stability of solutions to ordinary differential equations
34K05 General theory of functional-differential equations
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