ODE Solvers

# ODE Solvers

`solve(prob::ODEProblem,alg;kwargs)`

Solves the ODE defined by `prob` using the algorithm `alg`. If no algorithm is given, a default algorithm will be chosen.

## Recommended Methods

It is suggested that you try choosing an algorithm using the `alg_hints` keyword argument. However, in some cases you may want something specific, or you may just be curious. This guide is to help you choose the right algorithm.

### Non-Stiff Problems

For non-stiff problems, the native OrdinaryDiffEq.jl algorithms are vastly more efficient than the other choices. For most non-stiff problems, we recommend `Tsit5`. When more robust error control is required, `BS5` is a good choice. For fast solving at lower tolerances, we recommend `BS3`. For tolerances which are at about the truncation error of Float64 (1e-16), we recommend `Vern6`, `Vern7`, or `Vern8` as efficient choices.

For high accuracy non-stiff solving (BigFloat and tolerances like `<1e-20`), we recommend the `Feagin12` or `Feagin14` methods. These are more robust than Adams-Bashforth methods to discontinuities and achieve very high precision, and are much more efficient than the extrapolation methods. Note that the Feagin methods are the only high-order optimized methods which do not include a high-order interpolant (they do include a 3rd order Hermite interpolation if needed). If a high-order method is needed with a high order interpolant, then you should choose `Vern9` which is Order 9 with an Order 9 interpolant.

### Stiff Problems

For stiff problems at low tolerances it is recommended that you use `Rosenbrock23` As a native DifferentialEquations.jl solver, many Julia numeric types (such as Unitful, ArbFloats, or DecFP) will work. When the equation is defined via the `@ode_def` macro, this will be the most efficient. For faster solving when only the Jacobian is known and the macro is not used, use `radau`. High precision numbers are also compatible with `Trapezoid` which is a symplectic integrator. However, for the most efficient solvers for highly stiff equations which need high accuracy, use `radau` or `CVODE_BDF` provided by wrappers to the ODEInterface and Sundials packages respectively (see the conditional dependencies documentation). These algorithms require that the number types are Float64.

## Translations from MATLAB/Python/R

For users familiar with MATLAB/Python/R, good translations of the standard library methods are as follows:

• `ode23` –> `BS3()`

• `ode45`/`dopri5` –> `DP5()`, though in most cases `Tsit5()` is more efficient

• `ode23s` –> `Rosenbrock23()`

• `ode113` –> `CVODE_Adams()`, though in many cases `Vern7()` is more efficient

• `dop853` –> `DP8()`, though in most cases `Vern7()` is more efficient

• `ode15s`/`vode` –> `CVODE_BDF()`, though in many cases `radau()` is more efficient

• `ode23t` –> `Trapezoid()`

• `lsoda` –> `lsoda()` (requires `Pkg.add("LSODA"); using LSODA`)

• `ode15i` –> `IDA()`

# Full List of Methods

Choose one of these methods with the `alg` keyword in `solve`.

## OrdinaryDiffEq.jl

Unless otherwise specified, the OrdinaryDiffEq algorithms all come with a 3rd order Hermite polynomial interpolation. The algorithms denoted as having a "free" interpolation means that no extra steps are required for the interpolation. For the non-free higher order interpolating functions, the extra steps are computed lazily (i.e. not during the solve).

The OrdinaryDiffEq.jl algorithms achieve the highest performance for non-stiff equations while being the most generic: accepting the most Julia-based types, allow for sophisticated event handling, etc. They are recommended for all non-stiff problems. For stiff problems, the algorithms are currently not as high of order or as well-optimized as the ODEInterface.jl or Sundials.jl algorithms, and thus if the problem is on arrays of Float64, they are recommended. However, the stiff methods from OrdinaryDiffEq.jl are able to handle a larger generality of number types (arbitrary precision, etc.) and thus are recommended for stiff problems on non-Float64 numbers.

• `Euler`- The canonical forward Euler method.

• `Midpoint` - The second order midpoint method.

• `SSPRK22` - The two-stage, second order strong stability preserving (SSP) method of Shu and Osher. (free 2nd order SSP interpolant)

• `SSPRK33` - The three-stage, third order strong stability preserving (SSP) method of Shu and Osher. (free 2nd order SSP interpolant)

• `SSPRK432` - A 3/2 adaptive strong stability preserving (SSP) method with five stages. (free 2nd order SSP interpolant)

• `SSPRK104` - The ten-stage, fourth order strong stability preserving method of Ketcheson. (free 3rd order Hermite interpolant)

• `RK4` - The canonical Runge-Kutta Order 4 method.

• `BS3` - Bogacki-Shampine 3/2 method.

• `DP5` - Dormand-Prince's 5/4 Runge-Kutta method. (free 4th order interpolant)

• `Tsit5` - Tsitouras 5/4 Runge-Kutta method. (free 4th order interpolant)

• `BS5` - Bogacki-Shampine 5/4 Runge-Kutta method. (5th order interpolant)

• `Vern6` - Verner's "Most Efficient" 6/5 Runge-Kutta method. (6th order interpolant)

• `Vern7` - Verner's "Most Efficient" 7/6 Runge-Kutta method. (7th order interpolant)

• `TanYam7` - Tanaka-Yamashita 7 Runge-Kutta method.

• `DP8` - Hairer's 8/5/3 adaption of the Dormand-Prince 8 method Runge-Kutta method. (7th order interpolant)

• `TsitPap8` - Tsitouras-Papakostas 8/7 Runge-Kutta method.

• `Vern8` - Verner's "Most Efficient" 8/7 Runge-Kutta method. (8th order interpolant)

• `Vern9` - Verner's "Most Efficient" 9/8 Runge-Kutta method. (9th order interpolant)

• `Feagin10` - Feagin's 10th-order Runge-Kutta method.

• `Feagin12` - Feagin's 12th-order Runge-Kutta method.

• `Feagin14` - Feagin's 14th-order Runge-Kutta method.

• `ImplicitEuler` - A 1st order implicit solver. Unconditionally stable.

• `Trapezoid` - A second order unconditionally stable implicit solver. Good for highly stiff.

• `Rosenbrock23` - An Order 2/3 L-Stable fast solver which is good for mildy stiff equations with oscillations at low tolerances.

• `Rosenbrock32` - An Order 3/2 A-Stable fast solver which is good for mildy stiff equations without oscillations at low tolerances. Note that this method is prone to instability in the presence of oscillations, so use with caution.

Example usage:

``````alg = Tsit5()
solve(prob,alg)  ``````

### Extra Options

The following methods allow for specification of `linsolve`: the linear solver which is used:

• `Rosenbrock23`

• `Rosenbrock32`

For more information on specifying the linear solver, see the manual page on solver specification.

The following methods allow for specification of `nlsolve`: the nonlinear solver which is used:

• `ImplicitEuler`

• `Trapezoid`

Note that performance overload information (Jacobians etc.) are not used in this mode. For more information on specifying the nonlinear solver, see the manual page on solver specification.

Additionally, the following methods have extra differentiation controls:

• `Rosenbrock23`

• `Rosenbrock32`

• `ImplicitEuler`

• `Trapezoid`

In each of these, `autodiff` can be set to turn on/off autodifferentiation, and `chunk_size` can be used to set the chunksize of the Dual numbers (see the documentation for ForwardDiff.jl for details). In addition, `Rosenbrock23` and `Rosenbrock32` can set `diff_type`, which is the type of numerical differentiation that is used (when autodifferentiation is disabled). The choices are `:central` or `:forward`.

Examples:

``````sol = solve(prob,Rosenbrock23()) # Standard, uses autodiff
sol = solve(prob,Rosenbrock23(chunk_size=10)) # Autodiff with chunksize of 10
sol = solve(prob,Rosenbrock23(autodiff=false)) # Numerical differentiation with central differencing
sol = solve(prob,Rosenbrock23(autodiff=false,diff_type=:forward)) # Numerical differentiation with forward differencing``````

### Tableau Method

Additionally, there is the tableau method:

• `ExplicitRK` - A general Runge-Kutta solver which takes in a tableau. Can be adaptive. Tableaus

are specified via the keyword argument `tab=tableau`. The default tableau is for Dormand-Prince 4/5. Other supplied tableaus can be found in the Supplied Tableaus section.

Example usage:

``````alg = ExplicitRK(tableau=constructDormandPrince())
solve(prob,alg)``````

### CompositeAlgorithm

One unique feature of OrdinaryDiffEq.jl is the `CompositeAlgorithm`, which allows you to, with very minimal overhead, design a multimethod which switches between chosen algorithms as needed. The syntax is `CompositeAlgorthm(algtup,choice_function)` where `algtup` is a tuple of OrdinaryDiffEq.jl algorithms, and `choice_function` is a function which declares which method to use in the following step. For example, we can design a multimethod which uses `Tsit5()` but switches to `Vern7()` whenever `dt` is too small:

``````choice_function(integrator) = (Int(integrator.dt<0.001) + 1)
alg_switch = CompositeAlgorithm((Tsit5(),Vern7()),choice_function)``````

The `choice_function` takes in an `integrator` and thus all of the features available in the Integrator Interface can be used in the choice function.

## Sundials.jl

The Sundials suite is built around multistep methods. These methods are more efficient than other methods when the cost of the function calculations is really high, but for less costly functions the cost of nurturing the timestep overweighs the benefits. However, the BDF method is a classic method for stiff equations and "generally works".

• `CVODE_BDF` - CVode Backward Differentiation Formula (BDF) solver.

• `CVODE_Adams` - CVode Adams-Moulton solver.

The Sundials algorithms all come with a 3rd order Hermite polynomial interpolation. Note that the constructors for the Sundials algorithms take two main arguments:

• `method` - This is the method for solving the implicit equation. For BDF this defaults to `:Newton` while for Adams this defaults to `:Functional`. These choices match the recommended pairing in the Sundials.jl manual. However, note that using the `:Newton` method may take less iterations but requires more memory than the `:Function` iteration approach.

• `linearsolver` - This is the linear solver which is used in the `:Newton` method.

The choices for the linear solver are:

• `:Dense` - A dense linear solver.

• `:Band` - A solver specialized for banded Jacobians. If used, you must set the position of the upper and lower non-zero diagonals via `jac_upper` and `jac_lower`.

• `:Diagonal` - This method is specialized for diagonal Jacobians.

• `:BCG` - A Biconjugate gradient method.

• `:TFQMR` - A TFQMR method.

Example:

``````CVODE_BDF() # BDF method using Newton + Dense solver
CVODE_BDF(method=:Functional) # BDF method using Functional iterations
CVODE_BDF(linear_solver=:Band,jac_upper=3,jac_lower=3) # Banded solver with nonzero diagonals 3 up and 3 down
CVODE_BDF(linear_solver=:BCG) # Biconjugate gradient method                                   ``````

All of the additional options are available. The full constructor is:

``````CVODE_BDF(;method=:Newton,linear_solver=:Dense,
jac_upper=0,jac_lower=0,non_zero=0,krylov_dim=0,
stability_limit_detect=false,
max_hnil_warns = 10,
max_order = 5,
max_error_test_failures = 7,
max_nonlinear_iters = 3,
max_convergence_failures = 10)

jac_upper=0,jac_lower=0,krylov_dim=0,
stability_limit_detect=false,
max_hnil_warns = 10,
max_order = 12,
max_error_test_failures = 7,
max_nonlinear_iters = 3,
max_convergence_failures = 10)``````

See the Sundials manual for details on the additional options.

## ODE.jl

• `ode23` - Bogacki-Shampine's order 2/3 Runge-Kutta method

• `ode45` - A Dormand-Prince order 4/5 Runge-Kutta method

• `ode23s` - A modified Rosenbrock order 2/3 method due to Shampine

• `ode78` - A Fehlburg order 7/8 Runge-Kutta method

• `ode4` - The classic Runge-Kutta order 4 method

• `ode4ms` - A fixed-step, fixed order Adams-Bashforth-Moulton method†

• `ode4s` - A 4th order Rosenbrock method due to Shampine

†: Does not step to the interval endpoint. This can cause issues with discontinuity detection, and discrete variables need to be updated appropriately.

## ODEInterface.jl

The ODEInterface algorithms are the classic Hairer Fortran algorithms. While the non-stiff algorithms are superseded by the more featured and higher performance Julia implementations from OrdinaryDiffEq.jl, the stiff solvers such as `radau` are some of the most efficient methods available (but are restricted for use on arrays of Float64).

Note that this setup is not automatically included with DifferentialEquations.jl. To use the following algorithms, you must install and use ODEInterfaceDiffEq.jl:

``````Pkg.add("ODEInterfaceDiffEq")
using ODEInterfaceDiffEq``````
• `dopri5` - Hairer's classic implementation of the Dormand-Prince 4/5 method.

• `dop853` - Explicit Runge-Kutta 8(5,3) by Dormand-Prince.

• `odex` - GBS extrapolation-algorithm based on the midpoint rule.

• `seulex` - Extrapolation-algorithm based on the linear implicit Euler method.

• `radau` - Implicit Runge-Kutta (Radau IIA) of variable order between 5 and 13.

• `radau5` - Implicit Runge-Kutta method (Radau IIA) of order 5.

• `rodas` - Rosenbrock 4(3) method.

## LSODA.jl

This setup provides a wrapper to the algorithm LSODA, a well-known method which uses switching to solve both stiff and non-stiff equations.

• `lsoda` - The LSODA wrapper algorithm.

Note that this setup is not automatically included with DifferentialEquaitons.jl. To use the following algorithms, you must install and use LSODA.jl:

``````Pkg.add("LSODA")
using LSODA``````

## List of Supplied Tableaus

A large variety of tableaus have been supplied by default via DiffEqDevTools.jl. The list of tableaus can be found in the developer docs. For the most useful and common algorithms, a hand-optimized version is supplied in OrdinaryDiffEq.jl which is recommended for general uses (i.e. use `DP5` instead of `ExplicitRK` with `tableau=constructDormandPrince()`). However, these serve as a good method for comparing between tableaus and understanding the pros/cons of the methods. Implemented are every published tableau (that I know exists). Note that user-defined tableaus also are accepted. To see how to define a tableau, checkout the premade tableau source code. Tableau docstrings should have appropriate citations (if not, file an issue).

Plot recipes are provided which will plot the stability region for a given tableau.