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:

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

f should be specified as f(du,u,p,t) (or in-place as f(resid,du,u,p,t)). 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::DAEFunction,du0,u0,tspan,p=NullParameters();kwargs...)
  • DAEProblem{isinplace}(f,du0,u0,tspan,p=NullParameters();kwargs...) : Defines the DAE with the specified functions. isinplace optionally sets whether the function is inplace or not. This is determined automatically, but not inferred.

Parameters are optional, and if not given then a NullParameters() singleton will be used which will throw nice errors if you try to index non-existent parameters. Any extra keyword arguments are passed on to the solvers. For example, if you set a callback in the problem, then that callback will be added in every solve call.

For specifying Jacobians and mass matrices, see the DiffEqFunctions page.

Fields

  • f: The function in the ODE.
  • du0: The initial condition for the derivative.
  • u0: The initial condition.
  • tspan: The timespan for the problem.
  • 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.
  • p: The parameters for the problem. Defaults to NullParameters
  • kwargs: The keyword arguments passed onto the solves.

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:

#] add DiffEqProblemLibrary
using DiffEqProblemLibrary.DAEProblemLibrary
# load problems
DAEProblemLibrary.importdaeproblems()
prob = DAEProblemLibrary.prob_dae_resrob
sol = solve(prob,IDA())
DiffEqProblemLibrary.DAEProblemLibrary.prob_dae_resrobConstant

The Robertson biochemical reactions in DAE form

\[\frac{dy₁}{dt} = -k₁y₁+k₃y₂y₃\]
\[\frac{dy₂}{dt} = k₁y₁-k₂y₂^2-k₃y₂y₃\]
\[1 = y₁ + y₂ + y₃\]

where $k₁=0.04$, $k₂=3\times10^7$, $k₃=10^4$. For details, see: Hairer Norsett Wanner Solving Ordinary Differential Equations I - Nonstiff Problems Page 129 Usually solved on $[0,1e11]$