Directory structure

• bin contains the main executable after the successful compilation, contains also example input data files and user custom input files should be put here,
• CAPD contains some modules from the CAPD library version dated 2010-12-29 (excluding unused modules), tested to be compatible with the program package,
• config contains the compiler configuration files,
• include contains the source header files,
• obj contains the object files generated by gcc,
• src contains the source code files.

Compile instructions

Windows

• Install Dev-C++

  Download Dev-C++ 5.0 with Mingw/GCC and install it.

  I use default installation directory C:\DEV-CPP.
  If you choose another directory then you need to make appropriate changes in the next steps.

• Install Active Perl

  Download Active Perl Standard Distribution and install it.
  Again I choose default installation directory C:\Perl

• Add to PATH variable directories that contains g++, make and perl

  You need to add directories C:\dev-cpp\bin and C:\Perl\bin to the system variable PATH. It is needed to write shortly g++ instead of C:\dev-cpp\bin\g++.

  On Windows XP go to My Computer→Properties and on Advanced tab choose Environment variables. Then in system variables find and edit Path variable. Add C:\dev-cpp\bin; to be first directory on a path (semicolon is used to separate directories). Check if C:\perl\bin is already present, if not add also this directory.
  If you have no rights to add anything to system variables you can add to local user variable also named Path.

• Compile the program
• Download the program and unpack it
  

  The program package can be found at the location.

  Unpack the downloaded file into an empty directory (I will call it DIST_DIR, for example let DIST_DIR=E:\capd).

  To unpack archive simply double click on it to enter it and then copy and paste all files into empty directory.

• Compile the program
  

  Run terminal ( Start→Run… : cmd )

  Change directory to DIST_DIR and run mingw32-make target=win command

  			E:
  			cd capd
  			mingw32-make target=win	
  			

  Be patient, it can take some time depending on your computer configuration.

  Common errors:
  

  Name 'make'/'g++' is not recognized as internal or external command,…

  Type path on terminal and check that PATH contains C:\dev-cpp\bin directory if not go to step number 3

  There are many additional parameters that can be passed to make. They are explained in details in the readme file which comes with CAPD package.

Linux

• Requirements
  

  The CAPD library is almost independent on the other libraries. This means, that all the necessary libraries you will find in the standard repositories for most popular linux distributions. Use your package manager in your system and install:

  • gcc, g++
  • libX11-devel
  • make
• Compile the program
  

  Before you compile the program, make sure you have installed all the packages required for the version you want to built (as listed above). Assuming that the package was extracted into the ../capd directory, go to the subdirectory ../capd/make and call

  make target=X11

Running instructions

• Windows
  
change current directory into the bin directory. Under the command line ( Start→Run… : cmd ) write proof.exe "input data file name",
for example proof.exe paper_example.txt,
• Linux
  
change current directory into the bin directory. Under the terminal write ./proof "input data file name",
for example ./proof paper_example.txt,

Input file structure

• m - symmetric Galerkin projection dimension, the projection will contain modes a_k such that |k| <= m.
• nu - viscosity constant of the Burgers equation, should be positive, refer (1) in the article for the exact meaning. One may extend a single value to an interval by setting the next value nonzero.
• nu_interval_right_bound - a nonnegative value. The interval nu+[0,nu_interval_right_bound] defines a set of the viscosity constant values for which the algorithm will attempt to perform the proof. If one likes to obtain a result for a single value of the viscosity constant, this parameter should equal to 0.
• a_0 - value of the zero indexed mode (it is constant).
• s - order of polynomial decay of coefficients that is required from the constructed bounds and trapping regions, have to be an integer satisfying s ≥ 4.
• order - order of the Taylor method used by C0 Lohner algorithm.
• step - step size of the Taylor method used by C0 Lohner algorithm, a lot of attention should be given to the proper choice of the step size, especially at high nu values step size should be decreased, because the system is becoming stiff and blow ups may result.
• forcing perturbation diameter - diameter (diam) of the uniform forcing perturbation, namely the interval [-diam/2, diam/2] is added to each forcing mode appearing in the list that follows.
  
• forcing - list of the floating point numbers indicating forcing function that is used.
  Values corresponds to the modes in ascending order starting at a_1, each mode is defined by two consecutive values,
  first is the value of the real part of the mode, second is the value of the complex part of the mode. Number of elements in this list should be less or equal 2m.

The output

• proof_part1_the_trapping_region.txt this file contains a trapping region enclosing the fixed point, along with corresponding data (matrices, eigenvalues).
• proof_part2_exact_fixed_point_location.txt this file contains calculated exact rigorous fixed point location (self-consistent bounds containing the fixed point).
• proof_part3_the_l.txt this file contains the upper bound of the logarithmic norm on the trapping region.
• proof_part4_basin_of_attraction.txt This file contains data regarding the basin of attraction of the fixed point.
• a collection of files named n_0.txt, where n is a number. Each of those files stores the record from the rigorous integration of a set that has occurred when verifying if the all solutions in an absorbing set belongs to the basin of attraction of a fixed point.

License