### 5 unstable releases

0.3.0 | Mar 22, 2019 |
---|---|

0.2.0 | Feb 20, 2019 |

0.1.2 | Oct 20, 2018 |

0.1.1 | Sep 25, 2018 |

0.1.0 | Sep 11, 2018 |

#**318** in Math

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# ODE-solvers

Numerical methods to solve ordinary differential equations (ODEs) in Rust.

## Installation

To start using the crate in a project, the following dependency must be added in the project's Cargo.toml file:

`[`dependencies`]`
ode`-`solvers `=` `"`0.3.0`"`

Then, in the main file, add

`use` ode`-``solvers``::``*``;`

## Type alias definition

The numerical integration methods implemented in the crate support multi-dimensional systems. In order to define the dimension of the system, declare a type alias for the state vector. For instance

`type` `State` `=` `Vector3``<``f64``>``;`

The state representation of the system is based on the VectorN<T,D> structure defined in the nalgebra crate. For convenience, ode-solvers re-exports six types to work with systems of dimension 1 to 6: Vector1<T>,..., Vector6<T>. For higher dimensions, the user should import the nalgebra crate and define a VectorN<T,D> where the second type parameter of VectorN is a dimension name defined in nalgebra. Note that the type T must be f64. For instance, for a 9-dimensional system, one would have:

`type` `State` `=` `VectorN``<``f64`, `nalgebra``::`U9`>``;`

## System definition

The system of first order ODEs must be defined in the

method of the `system`

trait. Typically, this trait is defined for a structure containing some parameters of the model. The signature of the `System <V>`

`System``<`V`>`

trait is:`pub` `trait` `System`<V> `{`
`fn` `system``(``&``self`, `x``:` `f64`, `y``:` `&`V, `dy``:` `&``mut` V`)``;`
`fn` `solout``(``&``self`, `_x``:` `f64`, `_y``:` `&`V, `_dy``:` `&`V`)`` ``->` `bool` `{`
`false`
`}`
`}`

where

must contain the ODEs: the second argument is the independent variable (usually time), the third one is a vector containing the dependent variable(s), and the fourth one contains the derivative(s) of y with respect to x. The method `system`

is called after each successful integration step and stops the integration whenever it is evaluated as true. The implementation of that method is optional. See the examples for implementation details.`solout`

## Method selection

The following explicit Runge-Kutta methods are implemented in the current version of the crate:

Method | Name | Order | Error estimate order | Dense output order |
---|---|---|---|---|

Dormand-Prince | Dopri5 | 5 | 4 | 4 |

Dormand-Prince | Dop853 | 8 | (5, 3) | 7 |

These methods are defined in the modules dopri5 and dop853. The first step is to bring the desired module into scope:

`use` `ode_solvers``::``dopri5``::``*``;`

Then, a structure is created using the *new* or the *from_param* method of the corresponding struct. Refer to the API documentation for a description of the input arguments.

`let` `mut` stepper `=` `Dopri5``::`new`(`system`,` x0`,` x_end`,` dx`,` y0`,` rtol`,` atol`)``;`

The system is integrated using

`let` res `=` stepper`.``integrate``(``)``;`

and the results are retrieved with

`let` x_out `=` stepper`.``x_out``(``)``;`
`let` y_out `=` stepper`.``y_out``(``)``;`

See the homepage for more details.

## Acknowledgments

The algorithms implemented in this crate were originally implemented in FORTRAN by E. Hairer and G. Wanner, Université de Genève, Switzerland. This Rust implementation has been adapted from the C version written by J. Colinge, Université de Genève, Switzerland and the C++ version written by Blake Ashby, Stanford University, USA.

#### Dependencies

~3MB

~63K SLoC