Refined DAE Problems

# 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 SplitDAEProblem, 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.

### Constructors

SplitDAEProblem(f,u0,tspan,callback=nothing,mass_matrix=I) : Defines the ODE with the specified functions.

### Fields

• f: The tuple of functions in the ODE.

• u0: The initial condition.

• du0: The initial derivative condition.

• tspan: The timespan for the problem.

• callback: A callback to be applied to every solver which uses the problem. Defaults to nothing.

• mass_matrix: The mass-matrix. Defaults to I, the UniformScaling identity matrix.

## 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.

### Constructors

PartitionedDAEProblem(f,u0,tspan,callback=nothing,mass_matrix=I) : Defines the ODE with the specified functions.

### Fields

• f: The tuple of functions for the ODE.

• u0: The tuple of initial conditions.

• du0: The tuple of initial derivatives.

• tspan: The timespan for the problem.

• callback: A callback to be applied to every solver which uses the problem. Defaults to nothing.

• mass_matrix: The mass-matrix. Defaults to I, the UniformScaling identity matrix.