# The DiffEq Internals

The DiffEq solvers, OrdinaryDiffEq, StochasticDiffEq, FiniteElementDiffEq, etc. all follow a similar scheme which leads to rapid development and high performance. This portion of the documentation explains how the algorithms are written.

## Developing New Solver Algorithms

The easiest way to get started would be to add new solver algorithms. This is a pretty simple task as there are tools which put you right into the "hot loop". For example, take a look at the ODE solver code. The mode solve(::ODEProblem,::OrdinaryDiffEqAlgorithm) is glue code to a bunch of solver algorithms. The algorithms which are coded in DifferentialEquations.jl can be found in src/integrators.jl. The actual step is denoted by the perform_step!(integrator) function. For example, take a look at the Midpoint method's implementation:

@inline function perform_step!(integrator,cache::MidpointConstantCache,f=integrator.f)
@unpack t,dt,uprev,u,k = integrator
halfdt = dt/2
k = integrator.fsalfirst
k = f(t+halfdt,uprev+halfdt*k)
u = uprev + dt*k
integrator.fsallast = f(u,p,t+dt) # For interpolation, then FSAL'd
integrator.k = integrator.fsalfirst
integrator.k = integrator.fsallast
@pack integrator = t,dt,u
end

The available items are all unloaded from the integrator in the first line. fsalfirst inherits the value of fsallast on the last line. The algorithm is written in this form so that way the derivative of both endpoints is defined, allowing the vector integrator.k to determine a Hermite interpolation polynomial (in general, the k values for each algorithm form the interpolating polynomial). Other than that, the algorithm is as basic as it gets for the Midpoint method, making sure to set fsallast at the last line. The results are then packaged back into the integrator to mutate the state.

Note the name ConstantCache. In OrdinaryDiffEq.jl, the Algorithm types are used for holding type information about which solver to choose, and its the "cache" types which then hold the internal caches and state. ConstantCache types are for non in-place calculations f(u,p,t) and Cache types (like MidpointCache) include all of the internal arrays. Their constructor is specified in the cache.jl file. The cache (and the first fsalfirst) is initialized in the initialize! function next to the cache's perform_step! function.

The main inner loop can be summarized by the solve! command:

@inbounds while !isempty(integrator.opts.tstops)
while integrator.tdir*integrator.t < integrator.tdir*top(integrator.opts.tstops)
@ode_exit_conditions
perform_step!(integrator,integrator.cache)
loopfooter!(integrator)
if isempty(integrator.opts.tstops)
break
end
end
handle_tstop!(integrator)
end
postamble!(integrator)

The algorithm runs until tstop is empty. It hits the loopheader! in order to accept/reject the previous step and choose a new dt. This is done at the top so that way the iterator interface happens "mid-step" instead of "post-step". In here the algorithm is also chosen for the switching algorithms.

Then the perform_step! is called. (The exit conditions throw an error if necessary.) Afterwards, the loopfooter! is used to calculate new timesteps, save, and apply the callbacks. If a value of tstops is hit, the algorithm breaks out of the inner-most loop to save the value.

Adding algorithms to the other problems is very similar, just in a different package.

## Extras

If the method is a FSAL method then it needs to be set via isfsal and fsalfirst should be defined before the loop, with fsallast what's pushed up to fsalfirst upon a successful step. See :DP5 for an example.

If tests fail due to units (i.e. Unitful), don't worry. I would be willing to fix that up. To do so, you have to make sure you keep separate your rateTypes and your uTypes since the rates from f will have units of u but divided by a unit of time. If you simply try to write these into u, the units part will fail (normally you have to multiply by a $dt$).

If you want to access the value of u at the second-last time point, you can use uprev2 value. But in order to copy uprev value to uprev2 after each timestep, you need to make alg_extrapolates(alg::Your_Alg) = true in the alg_utils.jl file.