New solvers

[patrick-kidger]

SDE:

• [x] Milstein
• [x] Euler-Heun
• [x] SRKs (Additive)
• [ ] Talay (Ito commutative)

Implict:

• [ ] BDF
• [ ] Rosenbrock methods
• [x] IMEX methods, e.g. http://dx.doi.org/10.1175/mwr-d-13-00132.1

DAE:

• Lots of overlap with Rosenbrock methods, get them in at the same time.
• Standard DAE->ODE conversion (+optional nonlinear projections)
• Set made_jump=True if dconstraint/dz=0
• Mass matrices: work with these or work with constraint functions? Former is a little more general; maybe less efficient in some cases? Also think about traced mass matrices, possibly space-varying (e.g. quasilinear problems), possibly rank-varying.

Symplectic:

• Higher-order symplectic methods
• A lot of the lower-order methods (Stormer-Verlet, Leapfrog, Kick-drift-kick, Semi-implicit Euler) are all essentially the same thing. It would be good to write out something making this explicit, plus maybe add some special variants if you really want a particular variant.

Other:

• Higher-order ERK methods (Verner etc.)
• See also a few of the comments made about DifferentialEquations.jl in this (old) blog post: https://www.stochasticlifestyle.com/comparison-differential-equation-solver-suites-matlab-r-julia-python-c-fortran/

[trevormccrt]

@patrick-kidger are we still interested in Verlet methods (or other methods specifically taylored to Hamiltonian systems)?

[patrick-kidger]

Absolutely! I just haven't gotten around to implementing them. I'd be happy to take a PR on these.

[edoardoramalli]

@patrick-kidger I'm not able to properly initialise a KenCarp solver? Can you help me? Is something like ?

KenCarp4(MultiTerm([Bosh3, Kvaerno3]))

But when I use it to solve an ODE I got:

ValueError: terms must be a PyTree of AbstractTerms (such as ODETerm), with structure diffrax.term.MultiTerm[typing.Tuple[diffrax.term.AbstractTerm, diffrax.term.AbstractTerm]]

Can you please provide an entire example on how use the IMEX solvers?

[patrick-kidger]

For example:

explicit_term = ODETerm(lambda t, y, args: -y)
impliict_term = ODETerm(lambda t, y, args: -y)
terms = MultiTerm(explicit_term, implicit_term)
solver = KenCarp4()
diffeqsolve(terms, solver, ...)