Split ODE Problems


Defines a split ordinary differential equation (ODE) problem. Documentation Page: https://diffeq.sciml.ai/stable/types/splitodetypes/

Mathematical Specification of a Split ODE Problem

To define a SplitODEProblem, you simply need to give a two functions $f_1$ and $f_2$ along with an initial condition $u_0$ which define an ODE:

\[\frac{du}{dt} = f_1(u,p,t) + f_2(u,p,t)\]

f should be specified as f(u,p,t) (or in-place as f(du,u,p,t)), 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.

Many splits are at least partially linear. That is the equation:

\[\frac{du}{dt} = Au + f_2(u,p,t)\]

For how to define a linear function A, see the documentation for the DiffEqOperators.



The isinplace parameter can be omitted and will be determined using the signature of f2. Note that both f1 and f2 should support the in-place style if isinplace is true or they should both support the out-of-place style if isinplace is false. You cannot mix up the two styles.

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.

Under the hood, a SplitODEProblem is just a regular ODEProblem whose f is a SplitFunction. Therefore you can solve a SplitODEProblem using the same solvers for ODEProblem. For solvers dedicated to split problems, see Split ODE Solvers.

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


  • f1, f2: The functions in the ODE.
  • u0: The initial condition.
  • tspan: The timespan for the problem.
  • p: The parameters for the problem. Defaults to NullParameters
  • kwargs: The keyword arguments passed onto the solves.
SplitFunction{iip,F1,F2,TMM,C,Ta,Tt,TJ,JVP,VJP,JP,SP,TW,TWt,TPJ,S,O,TCV} <: AbstractODEFunction{iip}

A representation of a split ODE function f, defined by:

\[M \frac{du}{dt} = f_1(u,p,t) + f_2(u,p,t)\]

and all of its related functions, such as the Jacobian of f, its gradient with respect to time, and more. For all cases, u0 is the initial condition, p are the parameters, and t is the independent variable.

Generally, for ODE integrators the f_1 portion should be considered the "stiff portion of the model" with larger time scale separation, while the f_2 portion should be considered the "non-stiff portion". This interpretation is directly used in integrators like IMEX (implicit-explicit integrators) and exponential integrators.


                             mass_matrix = __has_mass_matrix(f) ? f.mass_matrix : I,
                             analytic = __has_analytic(f) ? f.analytic : nothing,
                             tgrad= __has_tgrad(f) ? f.tgrad : nothing,
                             jac = __has_jac(f) ? f.jac : nothing,
                             jvp = __has_jvp(f) ? f.jvp : nothing,
                             vjp = __has_vjp(f) ? f.vjp : nothing,
                             jac_prototype = __has_jac_prototype(f) ? f.jac_prototype : nothing,
                             sparsity = __has_sparsity(f) ? f.sparsity : jac_prototype,
                             paramjac = __has_paramjac(f) ? f.paramjac : nothing,
                             syms = __has_syms(f) ? f.syms : nothing,
                             indepsym= __has_indepsym(f) ? f.indepsym : nothing,
                             colorvec = __has_colorvec(f) ? f.colorvec : nothing,
                             sys = __has_sys(f) ? f.sys : nothing)

Note that only the functions f_i themselves are required. These functions should be given as f_i!(du,u,p,t) or du = f_i(u,p,t). See the section on iip for more details on in-place vs out-of-place handling.

All of the remaining functions are optional for improving or accelerating the usage of f. These include:

  • mass_matrix: the mass matrix M represented in the ODE function. Can be used to determine that the equation is actually a differential-algebraic equation (DAE) if M is singular. Note that in this case special solvers are required, see the DAE solver page for more details: https://diffeq.sciml.ai/stable/solvers/dae_solve/. Must be an AbstractArray or an AbstractSciMLOperator.
  • analytic(u0,p,t): used to pass an analytical solution function for the analytical solution of the ODE. Generally only used for testing and development of the solvers.
  • tgrad(dT,u,p,t) or dT=tgrad(u,p,t): returns $\frac{\partial f_1(u,p,t)}{\partial t}$
  • jac(J,u,p,t) or J=jac(u,p,t): returns $\frac{df_1}{du}$
  • jvp(Jv,v,u,p,t) or Jv=jvp(v,u,p,t): returns the directional derivative$\frac{df_1}{du} v$
  • vjp(Jv,v,u,p,t) or Jv=vjp(v,u,p,t): returns the adjoint derivative$\frac{df_1}{du}^\ast v$
  • jac_prototype: a prototype matrix matching the type that matches the Jacobian. For example, if the Jacobian is tridiagonal, then an appropriately sized Tridiagonal matrix can be used as the prototype and integrators will specialize on this structure where possible. Non-structured sparsity patterns should use a SparseMatrixCSC with a correct sparsity pattern for the Jacobian. The default is nothing, which means a dense Jacobian.
  • paramjac(pJ,u,p,t): returns the parameter Jacobian $\frac{df_1}{dp}$.
  • syms: the symbol names for the elements of the equation. This should match u0 in size. For example, if u0 = [0.0,1.0] and syms = [:x, :y], this will apply a canonical naming to the values, allowing sol[:x] in the solution and automatically naming values in plots.
  • indepsym: the canonical naming for the independent variable. Defaults to nothing, which internally uses t as the representation in any plots.
  • colorvec: a color vector according to the SparseDiffTools.jl definition for the sparsity pattern of the jac_prototype. This specializes the Jacobian construction when using finite differences and automatic differentiation to be computed in an accelerated manner based on the sparsity pattern. Defaults to nothing, which means a color vector will be internally computed on demand when required. The cost of this operation is highly dependent on the sparsity pattern.

Note on the Derivative Definition

The derivatives, such as the Jacobian, are only defined on the f1 portion of the split ODE. This is used to treat the f1 implicit while keeping the f2 portion explicit.

iip: In-Place vs Out-Of-Place

For more details on this argument, see the ODEFunction documentation.

recompile: Controlling Compilation and Specialization

For more details on this argument, see the ODEFunction documentation.


The fields of the SplitFunction type directly match the names of the inputs.

Symbolically Generating the Functions

See the modelingtoolkitize function from ModelingToolkit.jl for automatically symbolically generating the Jacobian and more from the numerically-defined functions. See ModelingToolkit.SplitODEProblem for information on generating the SplitFunction from this symbolic engine.

Solution Type

SplitODEProblem solutions return an ODESolution. For more information, see the ODE problem definition page for the ODESolution docstring.