Stability analysis of \(\theta\)-methods for neutral functional- differential equations. (English) Zbl 0824.65081

This paper deals with the numerical stability of the neutral functional differential equation \(y'(t) = ay(t) + by(qt) + cy'(pt)\), \(t > 0\). It is proved that \(\theta\)-methods are convergent if \(| c| < 1\). Experiments suggest they are divergent if \(| \theta|\) is large. The problem is transferred to a neutral equation with constant time lags. Using the later equation as a test model, it is shown that the linear \(\theta\)-method is \(\Lambda\)-stable if \(\text{Re }a < 0\) and \(| a| > | b|\) if and only if \(\theta \geq 1/2\) and the one-leg \(\theta\)-method is \(\Lambda\)-stable if \(\theta = 1\). It is also shown that inappropriate stepsize causes spurious solutions in the marginal case when \(\text{Re }a < 0\) and \(| a| = | b|\).


65L20 Stability and convergence of numerical methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
34K05 General theory of functional-differential equations
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