Question about DiffEqOperators

General Usage
1

I’m interested in possibly using the DiffEqOperators functionality in DifferentialEquations.jl

I usually solve systems of nonlinear equations that are of the form

\frac{\partial A}{\partial z} = i\beta(z) A + F(A,z)

where A and \beta are complex arrays (i.e. \beta is a linear operator), and F is some nonlinear function of A. The way I do this is to transform the above into

\frac{\partial G}{\partial z} = e^{-i\beta (z) z}F(A,z)

with

G = e^{-i\beta (z) z}A

And I have to transform back at each step to calculate F, and also transform back from G to A when saving the solution. Note that in general \beta depends on z, and so the operator needs to be updated as the integration progresses.

I currently solve the above by using the iterator interface to DifferentialEquations.jl, and mutating u after each step to reset it to A, before optionally saving and continuing. I get good performance from the Tsit5 algorithm, and I’m very happy (compared to when I was hand writing my own ODE solvers in C++).

However, I noticed the page on DiffEqOperators and wondered if that interface is designed to solve exactly this kind of problem?

Any tips?

2

Yes, it’s exactly for this kind of problem. More specifically, we have ExponentialUtilities.jl for doing the Krylov approximation to the matrix exponential

and then utilize this in the semilinear exponential integrators.

http://docs.juliadiffeq.org/latest/solvers/split_ode_solve.html#Semilinear-ODE-1

1 Like
3

Thanks!

Can I ask if these methods are efficient for diagonal operators (which is my case)? For example is the diagonal trait from LinearMaps.jl used? Regardless, I will give this a try.

Are there any examples anywhere of how to pull all of the pieces together?

Thanks!

4

If you make an array operator where it’s a Diagonal matrix then it will make use of the overloads. There is more we should be doing like this. We don’t have such a trait to generalize it to matrix-free operations yet or anything like that.

There’s some random tests around

But we really need to get a whole tutorial series out for it. There’s still some more on the API we want to settle first though.