Non-autonomous Linear ODE / Lie Group Problems

Non-autonomous linear ODEs show up in a lot of scientific problems where the differential equation lives on a manifold, such as a Lie Group. In these situations, specialized solvers can be utilized to enforce physical bounds on the solution and enhance the solving.

Mathematical Specification of a Non-autonomous Linear ODE

These algorithms require a Non-autonomous linear ODE of the form:

Where is an AbstractDiffEqOperator that is multiplied against . Many algorithms specialize on the form of , such as being a constant or being only time-dependent ( ).

Construction

Creating a non-autonomous linear ODE is the same as an ODEProblem, except f is represented by an AbstractDiffEqOperator (note: this means that any standard ODE solver can also be applied to problems written in this form). As an example:

function update_func(A, u, p, t)
    A[1, 1] = cos(t)
    A[2, 1] = sin(t)
    A[1, 2] = -sin(t)
    A[2, 2] = cos(t)
end
A = DiffEqArrayOperator(ones(2, 2), update_func = update_func)
prob = ODEProblem(A, ones(2), (10, 50.0))

defines a quasi-linear ODE where the components of are the given functions. Using that formulation, we can see that the general form is , for example:

function update_func(A, u, p, t)
    A[1, 1] = 0
    A[2, 1] = 1
    A[1, 2] = -2 * (1 - cos(u[2]) - u[2] * sin(u[2]))
    A[2, 2] = 0
end

has a state-dependent linear operator. Note that many other AbstractDiffEqOperators can be used, and DiffEqArrayOperator is just one version that represents A via a matrix (other choices are matrix-free).

Note that if is a constant, then it is sufficient to supply directly without an update_func.

Note About Affine Equations

Note that the affine equation

can be written as a linear form by extending the size of the system by one to have a constant term of 1. This is done by extending A with a new row, containing only zeros, and giving this new state an initial value of 1. Then extend A to have a new column containing the values of g(u,p,t). In this way, these types of equations can be handled by these specialized integrators.