DAE Problems

Mathematical Specification of an DAE Problem

To define a DAE Problem, you simply need to give the function $f$ and the initial condition $u₀$ which define an ODE:

\[0 = f(t,u,du)\]

f should be specified as f(t,u,du) (or in-place as f(t,u,du,resid)). Note that we are not limited to numbers or vectors for u₀; one is allowed to provide u₀ as arbitrary matrices / higher dimension tensors as well.

Problem Type

Constructors

DAEProblem(f,u0,du0,tspan) : Defines the ODE with the specified functions.

Fields

f: The function in the ODE.
u0: The initial condition.
du0: The initial condition for the derivative.
tspan: The timespan for the problem.
callback: A callback to be applied to every solver which uses the problem. Defaults to a black CallbackSet, which will have no effect.
differential_vars: A logical array which declares which variables are the differential (non algebraic) vars (i.e. du' is in the equations for this variable). Defaults to nothing. Some solvers may require this be set if an initial condition needs to be determined.

Refined DAE Problems

The refined DAE types are types that specify the DAE to a much greater degree of detail, and thus give the solver more information and make it easier to optimize. There are three different kinds of refined problems: split (IMEX) problems, partitioned problems, and constrained problems.

Mathematical Specification of a Split DAE Problem

To define a split DAEProblem, you simply need to give a tuple of functions $(f_1,f_2,\ldots,f_n)$ and the initial condition $u₀$ which define an ODE:

\[0 = f_1(t,u,u') + f_2(t,u,u') + \ldots + f_n(t,u,u')\]

f should be specified as f(t,u,du) (or in-place as f(t,u,du,res)), and u₀ should be an AbstractArray (or number) whose geometry matches the desired geometry of u.

Mathematical Specification of a Partitioned ODE Problem

To define a PartitionedDAEProblem, you need to give a tuple of functions $(f_1,f_2,\ldots,f_n)$ and the tuple of initial conditions $(u₀,v₀,...)$ (tuple of the same size) which define an ODE:

\[\frac{du}{dt} = f_1(t,u,v,...,du,dv,...) \\ \frac{dv}{dt} = f_2(t,u,v,...,du,dv,...) \\\]

f should be specified as f(t,u,v,...,du,dv,...) (or in-place as f(t,u,v,...,du,dv,...,res)), and the initial conditions should be AbstractArrays (or numbers) whose geometry matches the desired geometry of u. Note that we are not limited to numbers or vectors for u₀; one is allowed to provide u₀ as arbitrary matrices / higher dimension tensors as well.

Example Problems

Examples problems can be found in DiffEqProblemLibrary.jl.

To use a sample problem, such as prob_dae_resrob, you can do something like:

#Pkg.add("DiffEqProblemLibrary")
using DiffEqProblemLibrary
prob = prob_dae_resrob
sol = solve(prob,IDA())