DelayDiffEq: Generating Delay Differential Equation Solvers via Recursive Embedding of Ordinary Differential Equation Solvers

Abstract: Traditional solvers for delay differential equations (DDEs) are designed around only a single method and do not effectively use the infrastructure of their more-developed ordinary differential equation (ODE) counterparts. In this work we present DelayDiffEq, a Julia package for numerically solving delay differential equations (DDEs) which leverages the multitude of numerical algorithms in OrdinaryDiffEq for solving both stiff and non-stiff ODEs, and manages to solve challenging stiff DDEs. We describe how compiling the ODE integrator within itself, and accounting for discontinuity propagation, leads to a design that is effective for DDEs while using all of the ODE internals. We highlight some difficulties that a numerical DDE solver has to address, and explain how DelayDiffEq deals with these problems. We show how DelayDiffEq is able to solve difficult equations, how its stiff DDE solvers give efficiency on problems with time-scale separation, and how the design allows for generality and flexibility in usage such as being repurposed for generating solvers for stochastic delay differential equations.