Refined ODE Problems

# Refined ODE Problems

The refined ODE types are types that specify the ODE 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 ODE Problem

To define a SplitODEProblem, 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:

$\frac{du}{dt} = f_1(t,u) + f_2(t,u) + \ldots + f_n(t,u)$

f should be specified as f(t,u) (or in-place as f(t,u,du)), and u₀ should be an AbstractArray (or number) 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

SplitODEProblem(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.

• 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 PartitionedODEProblem, 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:

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

f should be specified as f(t,u,v,...) (or in-place as f(t,u,v,...,du)), 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. In some cases, the solvers may specify the functions in a split form, for example:

$\frac{du}{dt} = f_1(t,u,v,...) + f_2(t,u,v,...) \\ \frac{dv}{dt} = f_3(t,u,v,...) \\ ...$

See the solver's documentation for the form it is expecting.

### Constructors

PartitionedODEProblem(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.

• 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 an Second Order ODE Problem

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

$u'' = f(t,u,u')$

f should be specified as f(t,u,du) (or in-place as f(t,u,du,ddu)), and u₀ should be an AbstractArray (or number) 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.

From this form, a partitioned ODE

$u' = v \\ v' = f(t,u,v) \\$

is generated.

### Constructors

SecondOrderODEProblem(f,u0,du0,tspan,callback=CallbackSet(),mass_matrix=I) : Defines the ODE with the specified functions.

### Fields

• f: The function in the ODE.

• u0: The initial condition.

• du0: The initial derivative.

• 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 Constrained ODE Problem

The constrained ODE:

$\frac{du}{dt} = f(t,u,v) \\ 0 = g(t,u,v)$

is a type of refined ODE which specifies a DAE.

### Constructors

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

### Fields

• f: The tuple of functions for the ODE.

• g: The constraint equation.

• u0: The initial conditions.

• v0: The initial values for the purely-algebraic variables.

• 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 Split Constrained ODE Problem

The split constrained ODE:

$\frac{du}{dt} = f_1(t,u,v) + f_2(t,u,v) + ... + f_n(t,u,v) \\ 0 = g(t,u,v)$

is a type of refined ODE which specifies a DAE.

### Constructors

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

### Fields

• f: The tuple of functions for the ODE.

• g: The constraint equation.

• u0: The tuple of initial conditions.

• 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 Constrained ODE Problem

The constrained ODE:

$\frac{du}{dt} = f_1(t,u,v,...) \\ \frac{dv}{dt} = f_2(t,u,v,...) \\ ...\\ 0 = g(t,u,v)$

is a type of refined ODE which specifies a DAE. As with the standard Partitioned problem, the solver may specify a special form which may result in splitting ODEs in some of the systems.

### Constructors

ConstrainedODEProblem(f,u0,v0,tspan,callback=nothing,mass_matrix=I) : Defines the ODE with the specified functions.

### Fields

• f: The tuple of functions for the ODE.

• g: The constraint equation.

• u0: The tuple of initial conditions.

• v0: The tuple of initial values for the purely-algebraic variables.

• 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.