Explain irregular behavior of Algorithm 2 for the implicit delay–differential formulation

Determine the theoretical and numerical causes of the irregular behavior observed when applying Algorithm 2 (the solver based on the implicit delay–differential formulation given by α V_R(z_R(s), s) + γ = (√2/3) z_R′(s) and τ̂ ∂_s V_R = ∂_x^2 V_R − V_R + (1 + f_2(x)) sign(x − z_R(s)) ) to simulate front position dynamics. Specifically, ascertain why Algorithm 2 fails to match Algorithm 1 and does not produce periodic orbits in the Example 3 setting with localized heterogeneity f_2, and identify the mechanisms responsible for the observed discrepancies.

Background

The paper derives a delay–differential equation for the front position in a heterogeneous two-component reaction–diffusion system and proposes two numerical schemes: Algorithm 1 (a Monte Carlo approach using an explicit representation of the delay functional) and Algorithm 2 (a solver based on the implicit delay–PDE coupling for V_R and z_R). In Example 3, designed to test direction switching and periodic behavior induced by localized heterogeneities, Algorithm 2 did not match Algorithm 1 and failed to produce periodic orbits.

The authors explicitly note that the reasons for this discrepancy are currently unknown, highlighting a concrete unresolved issue about the validity, stability, or numerical consistency of the implicit solver relative to the explicit Monte Carlo approach. Clarifying this would help assess reliability of numerical reductions and guide future analytical paper of the delay–differential dynamics.