ODEInterface

This julia module provides an interface to solvers for ordinary differential equations (ODEs) written in Fortran for solving initial value problems (IVP) of the form

x' = rhs(t,x),      x(t₀) = x₀

or (for solvers supporting a "mass matrix" M)

M⋅x' = rhs(t,x),    x(t₀) = x₀.

Additionally a boundary value solver (called bvpsol) is supported for boundary value problems (BVP) of the form

x' = rhs(t,x),      r = bc( xa, xb ) = 0

This julia module does not contain code for solving initial value problems, but this module does contain code for interacting with compiled Fortran-solvers. That's the reason, why this module is not called ODESuite. See the requirements below how to get to the compiled Fortran-solvers.

Currently the following Fortran-solvers, written by Prof. E. Hairer and Prof. G. Wanner, are supported:

• dopri5: explicit Runge-Kutta method of order 5(4) due to Dormand & Prince
• dop853: explicit Runge-Kutta method of order 8(5,3) due to Dormand & Prince
• odex: GBS extrapolation-algorithm based on the explicit midpoint rule
• radau5: implicit Runge-Kutta method (Radau IIA) of order 5
• radau: implicit Runge-Kutta method (Radau IIA) of variable order between 5 and 13
• seulex: extrapolation-algorithm based on the linear implicit Euler method
• rodas: Rosenbrock method of order 4(3) (with possibly singular mass matrix)

see Software page of Prof. Hairer.

Additionally the following Fortran-solvers from the SLATEC Common Mathematical Library are supported:

• ddeabm: Adams-Bashforth-Moulton Predictor-Corrector method (order between 1 and 12)
• ddebdf: Backward Differentiation Formula (orders between 1 and 5)

Also supported:

• bvpsol: a boundary value problem solver for highly nonlinear two point boundary value problems using either a local linear solver or a global sparse linear solver. Please note: The license for bvpsol only covers non commercial use, see License. Written by P. Deuflhard, G. Bader, L. Weimann, see CodeLib at ZIB.
• colnew: a multi-point boundary value problem solver for mixed order systems using collocation. Written by U. Ascher, G. Bader, see Colnew Homepage
• BVP_M-2: a boundary value problem solver for the numerical solution of boundary value ordinary differential equations with defect and global error control. Written by J. J. Boisvert, P.H. Muir and R. J. Spiteri, see BVP_M-2 Page

Description: Calling the Solvers

The following features of the IVP-solvers are supported by this ODEInterface:

• providing an output function (e.g. for dense output or for event location) to the solvers
• providing mass- and jacobi-matrices for the solvers (with support for banded matrices)
• all the solvers' parameters for fine-tuning them, see Options for Solvers and Option Overview
• support for problems with "special structure", see special structure

This module needs the compiled Fortran solvers as shared libraries (i.e. .so, .dylib or .dll files, respectively). There are three ways to get them:

• The package ODEInterface_jll.jl has precompiled solvers for different platforms. Julia 1.3 or newer is needed for this. This is the default behaviour for julia versions 1.3 or newer.
• The build-script of this module: It tries to use gfortran and compile the solver libraries. This is the default behaviour for julia versions less than 1.3.
• You compile the solvers yourself (perhaps with different options and/or a different compiler). In this case just call ODEInterface.help_solversupport for further informations (help topics) on how to compile the solvers and how to create shared libraries.

Please see the help for ODEInterface.loadODESolves how to ignore the precompiled ODEInterface_jll package and use the/your locally built libraries.

see ODEInterface.help_overview for an overview of some help topics.

• basic examples can be found in examples/BasicExamples
• more advanced examples can be found in examples/AdvancedExamples
• the Reentry Problem of the Apollo11 mission can be found in examples/ReentryOfApollo11

The author of this julia module is

 Dr. Christian Ludwig
 email: ludwig@ma.tum.de
   (Faculty of Mathematics, Technische Universität München)