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 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:

\[u^\prime = A(u,p,t)u\]

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

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.))

defines a quasi-linear ODE $u^\prime = A(t)u$ where the components of $A$ are the given functions. Using that formulation, we can see that the general form is $u^\prime = A(u,p,t)u$, 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 $A$ is a constant, then it is sufficient to supply $A$ directly without an update_func.

Note About Affine Equations

Note that the affine equation

\[u^\prime = A(u,p,t)u + g(u,p,t)\]

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.