Operator Splitting and Implicit Euler Methods for Delay Differential Equations on Interpolation Spaces: Convergence Analysis in Autonomous and Non-Autonomous Sectorial and Non-Sectorial Regimes

TL;DR

This work develops a unified operator-theoretic framework for applying the Implicit Euler method and Lie-Trotter splitting to delay differential equations on a product space \mathcal{E}_0 = X \oplus L^1([\tau,0]). The generator is decomposed into a sectorial diffusion part, a left-shift transport part, and a delay coupling, enabling analysis in both autonomous and non-autonomous, sectorial and non-sectorial regimes. It derives first-order local defect estimates O($h$) with O($h^2$) time-dependent refinements, and global O($h$) convergence under power-bounded, Ritt, or Kreiss stability, by blending Lax–Richtmyer equivalence with Kato's evolution theory and interpolation spaces. Numerical experiments across four regimes validate the theory and demonstrate that Lie-Trotter splitting delivers competitive accuracy with substantially reduced cost, especially in non-autonomous sectorial cases where full inversions are avoided. The results forge a concrete link between abstract semigroup theory and practical numerical schemes for delay equations, with clear guidance for implementing efficient, reliable splitting methods in complex delay PDEs.

Abstract

We study the convergence of resolvent-based Lie-Trotter splitting schemes for delay differential and delay partial differential equations formulated on product spaces of the form $\mathcal{E}_0=X \oplus L^1([τ, 0])$. The generator is decomposed into a sectorial diffusion part, a transport part generating the left-shift semigroup, and a bounded (or $C$-bounded) delay coupling. For both autonomous and non-autonomous systems we derive uniform local defect estimates of order $O(h)$, complemented by $O\left(h^2\right)$ corrections when coefficients vary in time. Global convergence of order $O(h)$ follows under power-bounded, Ritt, or Kreiss stability. The analysis combines the Lax-Richtmyer equivalence principle for operator splittings with Kato's theory of non-autonomous evolution equations, thus unifying sectorial and non-sectorial settings within a single perturbation framework. Finally, we perform numerical calculations for a specific DDE to verify the effectiveness of the Lie-Trotter operator splitting.

Figures (15)

• Figure 1:
• Figure 3: (Left) $\log\|u(t)\|$ vs. $t$, (Right) Relative Error $\|u(t)\|/Me^{\omega t}$, $a=-0.15$, $b=-6.0$, $\tau=-8.0$, history = 10th Polynomial.
• Figure 4:
• Figure 6:
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Theorems & Definitions (86)

• Definition 2.1: Sectorial operator (Lunardi)
• Definition 2.2: Analytic semigroup (Lunardi)
• Definition 2.3: $C$--bounded operators
• Definition 2.4: Lipschitz continuity in operator norm
• Proposition 3.1: Shift generator and bounded perturbation on $\mathcal{D}(C)$
• Proposition 3.2: Local defect on interpolation scales
• proof : Idea of proof
• Remark 3.3: Alternative realization via free shift
• Lemma 3.4: closedness and density of the transport/shift generator
• proof