On the P-Stability of One-Step Collocation for Delay Differential Equations

In this paper we present a stability analysis of the one-step collocation method at Gaussian points for DDEs, based on the test equation $${\rm{y}}\prime\left({\rm{t}}\right) ={\rm{ay}}\left({\rm{t}}\right)+{\rm{by}}\left({{\rm{t}}-\tau}\right),{\rm{t}}>0$$ $${\rm{y}}\left( {\rm{t}}\right)=\phi\left( {\rm{t}}\right){\rm{for}}-\tau\mathbin{\lower.3ex\hbox{$\buildrel<\over{\smash{\scriptstyle=}\vphantom{_x}}$}}{\rm{t}}\mathbin{\lower.3ex\hbox{$\buildrel<\over{\smash{\scriptstyle=}\vphantom{_x}}$}}0$$ where a and b are complex coefficients and τ is any positive constant delay. Convergence and superconvergence results for this method were recently proved. We show that it is P-stable, that is it yields an approximate solution u which, as y itself does, asymptotically vanishes if ∣b∣<−Re(a).

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