An Asynchronous multi-rate Taylor method for Delay Differential Equations

Abstract

The numerical simulation of high-dimensional, multi-rate Delay Differential Equations (DDEs) is fundamentally bottlenecked by synchronous time-stepping and the dynamic memory allocation required for continuous history tracking. In this paper, we introduce the Asynchronous Adaptive Taylor Solver (AATS), an event-driven integration framework designed to overcome these high-performance computing limitations. By assigning independent local clocks to individual coordinates and advancing them using high-order Taylor polynomials generated via compile-time Automatic Differentiation, AATS restricts computational work to actively evolving sub-graphs. To eliminate the severe memory overhead endemic to traditional DDE solvers, AATS utilizes statically allocated circular buffers to store polynomial segments, achieving interpolation-free continuous dense-output evaluation with a verified zero-allocation runtime memory footprint. Alongside this software architecture, we establish a novel continuous proof of convergence for asynchronous Taylor expansions and formally prove that the framework's algorithmic complexity scales linearly (O(N)). Extensive benchmarks against state-of-the-art synchronous solvers (Julia SciML) validate these theoretical bounds. On large-scale benchmarks (upto $N = 10000$ coordinates) AATS fundamentally minimizes the constant factor of algorithmic work by avoiding redundant evaluations, delivering empirically consistent with O(N) execution scaling and significant wall-clock speedups.