Delay-dependent stability of symmetric schemes in Boundary Value Methods for DDEs

We consider the symmetric schemes in Boundary Value Methods (BVMs) applied to delay differential equations y′(t)=ay(t)+by(t-τ)y′(t)=ay(t)+by(t-τ) with real coefficients a and b  . If the numerical solution tends to zero whenever the exact solution does, the symmetric scheme with (k1+m,k2)(k1+m,k2)-boundary conditions is called τk1,k2(0)τk1,k2(0)-stable. Three families of symmetric schemes, namely the Extended Trapezoidal Rules of first (ETRs) and second (ETR2s) kind, and the Top Order Methods (TOMs), are considered in this paper.By using the boundary locus technology, the delay-dependent stability region of the symmetric schemes are analyzed and their boundaries are found. Then by using a necessary and sufficient condition, the considered symmetric schemes are proved to be τν,ν-1(0)τν,ν-1(0)-stable.