# DDE Solvers

`solve(prob::AbstractDDEProblem, alg; kwargs)`

Solves the DDE defined by `prob`

using the algorithm `alg`

. If no algorithm is given, a default algorithm will be chosen.

## Recommended Methods

The recommended method for DDE problems are the `MethodOfSteps`

algorithms. These are constructed from an OrdinaryDiffEq.jl algorithm as follows:

`MethodOfSteps(alg; constrained = false, fpsolve = FPFunctional(; max_iter = 10))`

where `alg`

is an OrdinaryDiffEq.jl algorithm. Most algorithms should work.

### Nonstiff DDEs

The standard algorithm choice is `MethodOfSteps(Tsit5())`

. This is a highly efficient FSAL 5th order algorithm with free interpolants which should handle most problems. For fast solving where non-strict error control is needed, choosing `MethodOfSteps(BS3())`

can do well. Using `BS3`

is similar to the MATLAB `dde23`

. For algorithms where strict error control is needed, it is recommended that one uses `MethodOfSteps(Vern6())`

. Benchmarks show that going to higher order methods like `MethodOfSteps(DP8())`

may not be beneficial.

### Stiff DDEs and Differential-Algebraic Delay Equations (DADEs)

For stiff DDEs, the SDIRK and Rosenbrock methods are very efficient as they will reuse the Jacobian in the unconstrained stepping iterations. One should choose from the methods which have stiff-aware interpolants for better stability. `MethodOfSteps(Rosenbrock23())`

is a good low order method choice. Additionally, the `Rodas`

methods like `MethodOfSteps(Rodas4())`

are good choices because of their higher order stiff-aware interpolant.

Additionally, DADEs can be solved by specifying the problem in mass matrix form. The Rosenbrock methods are good choices in these situations.

### Lag Handling

Lags are declared separately from their use. One can use any lag by simply using the interpolant of `h`

at that point. However, one should use caution in order to achieve the best accuracy. When lags are declared, the solvers can more efficiently be more accurate. Constant delays are propagated until the order is higher than the order of the integrator. If state-dependent delays are declared, the algorithm will detect discontinuities arising from these delays and adjust the step size such that these discontinuities are included in the mesh, if steps are rejected. This way, all discontinuities are treated exactly.

If there are undeclared lags, the discontinuities due to delays are not tracked. In this case, one should only use residual control methods like `MethodOfSteps(RK4())`

, which is the current best choice, as these will step more accurately. Still, residual control is an error-prone method. We recommend setting the tolerances lower in order to get accurate results, though this may be costly since it will use a rejection-based approach to adapt to the delay discontinuities.

## Special Keyword Arguments

`discontinuity_interp_points`

- Number of interpolation points used to track discontinuities arising from dependent delays. Defaults to 10. Only relevant if dependent delays are declared.`discontinuity_abstol`

and`discontinuity_reltol`

- These are absolute and relative tolerances used by the check whether the time point at the beginning of the current step is a discontinuity arising from dependent delays. Defaults to 1/10^12 and 0. Only relevant if dependent delays are declared.

### Note

If the method is having trouble, one may want to adjust the fixed-point iteration. Decreasing the absolute tolerance and the relative tolerance by specifying the keyword arguments `abstol`

and `reltol`

when solving the DDE problem, and increasing the maximal number of iterations by specifying the keyword argument `max_iter`

in the `MethodOfSteps`

algorithm, can help ensure that the steps are correct. If the problem still is not correctly converging, one should lower `dtmax`

. For problems with only constant delays, in the worst case scenario, one may need to set `constrained = true`

which will constrain timesteps to at most the size of the minimal lag and hence forces more stability at the cost of smaller timesteps.