MATLAB Toolbox

The indigo MATLAB toolbox collects multiple integrators for differential and differential-algebraic equations (ODEs/DAEs). The toolbox provides an efficient interface for the integration of dynamical systems, which allows to easily switch between different integration methods without changing the code of the system of ODEs/DAEs. The toolbox is structured through a set of classes and functions which are described in the following sections.

Installation

Before talking about the structure of the toolbox, let’s see how to install it. There are multiple ways to install the toolbox.

  • The easiest way is to download the latest toolbox release from the repository releases section. In this case, the package is a compressed in a .zip file format. Unzip the file and add the folder to the MATLAB path using the setup.m file.

  • Analogously the latest development version can be downloaded from the GitHub repository and add the folder to the MATLAB path using the setup.m file. If you want to install the MATLAB toolbox system-wide, compile the MATLAB toolbox using the project file Indigo.prj in the matlab folder of the repository. The complied toolbox will be saved in the file indigo.mtlbx. Open the indigo.mtlbx file and follow the instructions.

  • You can also install the toolbox directly from the Add-On Explorer by clicking on the Get Add-Ons button in the Add-On Explorer tab. Then search for indigo and install the toolbox.

Structure

The toolbox is structured into three sections: the first section contains the Runge-Kutta solvers, the second section contains the types of ODEs/DAEs systems that can be integrated, and the third section contains the utilities functions used by the solvers.

The Runge-Kutta are contained in the folder indigo/RungeKutta, and implements into the following classes:

  • ODEsolver: (Abstract) class container for solvers of the system of Ordinary Differential Equations (ODEs) or Differential Algebraic Equations (DAEs).

  • RKexplicit: (Abstract) class implementing the function step for the explicit (and explicit-embedded) Runge-Kutta integration method.

  • RKimplicit: (Abstract) class implementing the function step for the implicit (and implicit-embedded) Runge-Kutta integration method.

The types of ODEs/DAEs systems, which are contained in the folder indigo/ODEsystem, and are implements into the following classes:

  • Base: (Abstract) class container for the system of Ordinary Differential Equations (ODEs) or Differential Algebraic Equations (DAEs).

  • ODEsystem: (Abstract) class container for a system of Ordinary Differential Equations (ODEs).

The utilities functions, which are contained in the folder indigo/Utils, and are implements into the following functions:

  • NewtonSolver: function implementing a Newton solver with affine invariant step for the explicit and implicit Runge-Kutta solvers.

  • SavePNG: function for saving the current figure as a PNG file.

    Available explicit Runge-Kutta solvers

    ExplicitEuler

    ExplicitMidpoint

    Generic2

    Generic3

    Heun2

    Heun3

    Ralston2

    Ralston3

    Ralston4

    RK3

    RK4

    RK38

    SSPRK3

    Wray3

    -

    -
    Available implicit Runge-Kutta solvers

    CrankNicolson

    GaussLegendre2

    GaussLegendre4

    GaussLegendre6

    ImplicitEuler

    ImplicitMidpoint

    LobattoIIIA2

    LobattoIIIA4

    LobattoIIIB2

    LobattoIIIB4

    LobattoIIIC2

    LobattoIIIC4

    LobattoIIICS2

    LobattoIIICS4

    LobattoIIID2

    LobattoIIID4

    RadauIA3

    RadauIA5

    RadauIIA3

    RadauIIA5

    SunGeng5

    -

    -

    -
    Available embedded explicit Runge-Kutta solvers

    BogackiShampine23

    CashKarp45

    DormandPrince45

    Fehlberg12

    Fehlberg45I

    Fehlberg45II

    Fehlberg78

    HeunEuler21

    Merson45

    Verner65

    Zonnenveld45

    -
    Available embedded implicit Runge-Kutta solvers

    GaussLegendre34

    GaussLegendre56

    LobattoIIIA12

    LobattoIIIA34

    LobattoIIIB12

    LobattoIIIB34

    LobattoIIIC12

    LobattoIIIC34

Usage

To use the toolbox you just have to follow the following steps.

  1. Create an instance of the class representing the system of ODEs/DAEs, which must be a subclass of the class ODEsystem;

  2. Create an instance of the class representing the solver, by simply instantiating one of the available solver classes listed in the tables above.

  3. Associate the system of ODEs/DAEs to the solver instance by calling the method set_ode of the solver instance. Finally, call the method solve of the solver instance can be called to solve the system of ODEs/DAEs. It will return the solution of the system of ODEs/DAEs as a matrix, where each column is the solution of the system at a different time instant.

  4. (Optional) The method plot of the solver instance can be called to plot the solution of the system of ODEs/DAEs.

  5. (Optional) The method animate of the solver instance can be called to animate the solution of the system of ODEs/DAEs.

For further details, please refer to the documentation of the classes and functions. Analogously, the examples provided in the tests section can be used as a starting point for the usage of the indigo toolbox.