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) 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)-boundary conditions is called τ_k1,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 τ_ν,ν-1(0)-stable.