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README
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CentPack Documentation v1.0.5
April 15, 2010
Copyright (C) 2010 Jorge Balbás and Eitan Tadmor
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TABLE OF CONTENTS
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I.
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Brief Description of Software and Algorithm
The files in this distribution implement six different non-oscillatory central schemes for solving hyperbolic systems of PDEs in one-,
u_t + f(u)_x = 0, (1)
and two-space dimensions,
u_t + f(u)_x + g(u)_y =0, (2)
over the domain [x_init, x_final] x [y_init, y_final], from time t=0 to t=t_final.
The algorithm is based on the evolution of the cell averages of u, and it is implemented in two main steps: #1 reconstruction of point values from cell averages and #2 evolution.
Although only a minimal number of auxiliary functions containing model and problem specific information (i.e., fluxes, initial and boundary conditions, output variables, etc.) are required from the user (examples are provided), and few (if any) modifications of the core files should be needed to solve most problems that accept the above formulations, (1) or (2), the source code can be easily modified and adapted to the taste and requirements of its user.
For more information about central schemes and their implementation consult the additional documentation and references therein or visit CentPack's website
/centpack/
For additional information about the variables used by each subroutine, their input, and/or output, please refer to the comments provided within each file and/or the guide CP_user_guide.pdf provided with this distribution.
This package (or previous versions of it) has been compiled, tested and validated in a variety of UNIX / UNIX-like platforms (e.g., Solaris, Linux, Mac OS/X) for a variety of test problems corresponding to different models (e.g., Euler equations of gas dynamics and Ideal MHD equations, Shallow-water equations). The compilation and execution instructions provided below and the included Makefiles are linux-based (g++ compiler), but they should work on other platforms with minor or no modifications.
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II.
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Download
Pre-compiled CentPack binaries for different architectures are available at
/centpack/software
A full source code distiribution is also available for downlod for registered users. The source code version will allow users to modify and adapt CentPack best for their specific needs.
Both distributions --precompiled binaries and full source code alike-- are distributed with additional source files that allow users to compile and run sample applications.
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III.
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Description of this Package
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CentPack C++ libraries
The core of this software consists of seven C++ libraries and their corresponding header files. The first one, libarray.a, implement the class of multidimensional dynamic data structures (arrays) used to store the solution (and other intermediate and associated) variables. The other six libraries, libcentpack_Xd_YDZ.a, implement different versions of non-oscillatory central schemes according to the following notation:
X = number of space dimensions: X = 1 or 2,
YD = formulation: Y = F for Fully-Discrete or Y = S for Semi-Discrete, and
Z = order of accuracy of the scheme: 2 for 2nd order, and 3 for 3rd.
For example, the library libcentpack_1d_SD2.a implements a one-dimensional, semi-discrete, 2nd order central solver.
REMARKS:
| 1. |
The source code distribution contains the source and make files needed to create this libraries, not the libraries themselves. For a detailed description of each subroutine, please refer to the file describing the subroutine and/or the manual CP_user_guide.pdf.
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| 2. |
The number of source files compiled under each library depends on the dimensions, formulation, and order of accuracy of the specific central scheme implemented by that library.
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| 3. |
The fully-discrete solvers provided in this package are only 2nd order. |
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Model Specific Files (user specified, examples provided)
| 1. |
fluxx.cc -- defines the function f(u) in equation (1). |
| 2. |
fluxy.cc -- defines the function g(u) in equation (1). THIS FUNCTION IS ONLY REQUIRED FOR 2d SOLVERS!
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| 3. |
spectral_radii.cc -- estimates the maximum speed of propagation of the components of u in both space dimensions. FOR 1d SOLVERS THE NAME OF THIS FUNCTION IS spectral_radius.cc.
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| 4. |
writeout.cc -- defines the output variables as two-dimensional arrays and a rule to extract them from the multi-dimensional array holding the solution u of equation (1) or (2), the variables are extracted and written to files adequately numbered for further manipulation and analysis of results (e.g., plotting, error analysis, etc.).
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| C. |
Problem Specific Files (user specified, examples provided)
| 1. |
initial_conditions.cc -- defines a rule to define the solution mesh and initialize the array holding the solution u. The function is called by CentPack's main routine before initiating the evolution.
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boundary_conditions.cc -- boundary conditions are implemented by applying them to four "ghost" rows and columns (two around each boundary of the solution domain) in the solution array u. This function provides with a rule to fill those rows/columns according to the problem boundary conditions. This allows for the computation of the solution over the computational domain can be carried out
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| 3. |
input -- a short file containing the required input required by CentPack to compute the solution (i.e., number of grid cells along each dimension, cfl restriction, final time of simulation, and other model/problem specific information); the value for each input variable is written on a separate line of the file in the following order (those with addiTional information in parenthesis, should be omitted in input files for other solvers):
x_init -- left end-point of domain in x-direction
x_final -- right end-point of domain in x-direction
y_init -- left end-point of domain in y-direction (2d solvers only)
y_final -- right end-point of domain in y-direction (2d solvers only)
J -- number of grid cells in x-direction
K -- number of grid cells in y-direction (2d solvers only)
L -- number of components of u
gamma -- parameter specifying constant ratio of specific heats
t_final -- final time of simulation
dt_out -- time interval at which intermediate output is desired
cfl -- time step restriction
alpha -- parameter for second order reconstruction (still required as input for 3rd order)
B1 -- free parameter (from MHD magnetic field, only required for 1d solvers)
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Make Files
CentPack is distributed with a number of make files that allow the user to compile and install the libraries described above as well as several examples. When executed, and according to the additional arguments given with the make command, the Makefile at the top CentPack directory, calls recursively to other make files in the src/ and src/Xd/samples/ (X = 1,2) subdirectories. These files can be easily modified in order for users to compile their own central solvers.
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MATLAB Files
The examples provided with the source code in
src/Xd/samples/EXAMPLE_NAME
(X = 1,2), include an MATLAB file, EXAMPLE_NAME_frames.m (_frames actually
replaces the _YDZ extension in EXAMPLE_NAME), that creates plots of the solution
variables.
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IV. |
Installation Instructions
| A. |
REMARKS and NOTATION
| 1. |
The following instructions assume that the user specified files have been created (or the provided examples are used).
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| 2. |
The > symbol below represents the command line prompt of the command shell.
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The quotation marks below are used to refer to specific file names, they are not part of the name or needed in the command line syntax.
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The files named "Makefile" contain compilation instructions for the g++ compiler, adjust as needed for your C++ compiler.
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| B. |
How to Install CentPack
If you downloaded pre-compiled binaries, you can skip to IV.C below. If you downloaded CentPack's source code:
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Move or copy the downloaded archive CP-1-0.tar.gz to your home directory (or anywhere else in your hard drive where you have write permission), and cd there
> cp CP-1-0.tar.gz $HOME/CP-1-0.tar.gz
> cd $HOME
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Extract the contents of the archive (a directory called CP-1.0) and cd to CP-1.0
> tar -zxf CP-1-0.tar.gz
> cd CP-1.0
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| 3. |
To compile and install CentPack type
> make libs
This will create seven libraries in the subdirectory CP-1.0/lib/, one with the definition of the array classes (data structures) used by CentPack, and six with the implementation of each one of the central solvers implemented by this version of CentPack. You can verify this by typing
> ls lib/
if everything went well, you should see a file called liabarray.a, and six other that follow the notation libcentpack_Xd_YDZ.a, with X, Y, and Z, taking on the values specified above (section III.A).
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| C. |
How to Compile ALL examples
Make sure you are in CP-1.0, and type
> make samples
This will create create a new directory, CP-1.0/samples/, containing six subdirectories consisting of the name of the example, followed by a key of the form _Xd_YDZ denoting the central solver used to compile the example as specified above (e.g., burguers_1d_SD3, euler_2d_FD2, etc.) These directories contain everything needed to run the example and reproduce the animations shown in CentPack's sample pages.
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How to compile a single example
Alternatively, individual examples can be compiled by typing
> make NAME_OF_EXAMPLE
with NAME_OF_EXAMPLE substituted by one of the following:
burgers_1d_SD3, euler_1d_SD2, MHD_1d_FD2, euler_2d_FD2, MHD_2d_SD2, and scalar_2d_SD3. For example, the command
> make burgers_1d_SD3
will create the directory CP-1.0/samples/burgers_1d_SD3/ with all the files needed to run the simulation and directories to write the output.
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V. |
Running a CentPack solver
| A. |
Running one of CentPack's examples (assuming burgers_1d_SD3 as the example to be run)
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After compiling the example(s), go into the sample directory you wish to run. Make sure you are in CP-1.0, and type
> cd samples/burgers_1d_SD3/
Alternatively, if you compiled other examples, you can browse CP-1.0/samples/ and decide which example you want to run, then move to its simulation directory, for example
> ls samples/
> cd samples/euler_1d_SD2/
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Once in the directory of the example you wish to run, type its name, in our example,
> ./burgers_1d_SD3
Note that each simulation directory contains only one executable whose name is quite obvious, the only other contents are the file named 'input', and few directories ending in _files/, where CentPack's output will be written.
This command will start running CentPack's solver. CentPack will read the data in the input file, set the initial conditions, and evolve them according to the hyperbolic conservation law, equation (1). After each time iteration, CentPack will write a line to the screen that will help monitor the simulation. This line contains the following information:
run -- present simulation time
dt -- time step used in the last time iteration
dt_cpu -- CPU time employed in last time iteration
t -- total CPU time thus far
odd/even -- (fully-discrete schemes only) indicates whether the last time step was an even or odd one
Alternatively, you can run CentPack's example by typing
> ./burgers_1d_SD3 > monitor &
This will re-direct the monitoring information to the file monitor.
Depending in your user privileges, the performance of the solver can be improved by typing
> nice ./burgers_1d_SD3 > monitor &
When the simulation is completed, in addition to the output files, a file named run_info.txt containing some simulation information, will be generated |
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| B. |
CentPack's Output (assuming burgers_1d_SD3 is the example run)
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The parameter dt_out, provided in the input file, will determine at what approximate time interval the output will be produced. CentPack's output is produced by the function writeout.cc (examples of this function are provided), which specifies the names and of the variables to be output and a rule to calculate them. Note that this output variables are not necessarily the ones evolved by CentPack, e.g., when solving Euler's equations of gas dynamics, one may want to print out the pressure and not the conserved energy.
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In our example, the output produced by CentPack is written to two subdirectories of burgers_1d_SD3: u_files/ and t_files/. The first contains files with the numerical solution of equation (1), u_0, u_1, u_2, etc. (with u_0 corresponding to the initial conditions). The directory t_files/ contains the files t_0, t_1, t_2, etc., each holding a single value indicating the simulation time at which the corresponding u_n file was written. |
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What to do with CentPack's Output
| 1. |
Each of the examples provided include a MATLAB file,
EXAMPLE_NAME_frames.m. This .m file is copied into the simulation directory
of each example and when run from MATLAB, it will create and store the plots of the solution variables of the corresponding example. Type
>> help burgers_1d_frames
in MATLAB for detail information on how to use these files and instructions to create a gif animation. |
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The output files can be easily loaded into MATLAB (C) or similar software applications for scientific computing (e.g., Scilab (C) or Octave (C)) and manipulated with those applications for further analysis of the results.
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The output function does not write the mesh coordinates, x and y. These are not time dependent (i.e., there is no need to output them every single time the output function is called) and can be easily generated by any of the applications mentioned above.
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VI.
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Additional Remarks
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Additional Calculations
Some hyperbolic models may require the enforcement of additional constraints or, in some cases, the user may want to compute additional information related to the problem, but not explicitly calculated in the evolution of the conservation law. The code is written so as to allow such modifications as part of the existing routines or as new routines that can be added without altering the original structure of the package. Many such additions have been successfully created and tested by the authors before, some examples include the computation of the transverse kinetic energy or the addition of a external routines to enforce the solenoidal condition of the magnetic field in MHD test problems.
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Boundary rows and columns
The solution variables declared in the source files are of size (J+4) x (K+4), the values of the solution over the discretized domain are calculated and stored in the interior rows and columns (those with indexes ranging form 2 to J+1 and 2 to K+1 respectively). The additional rows and columns (rows numbered 0, 1, J+2, and J+3 and columns numbered 0, 1, K+2, and K+3) are filled/calculated according to the boundary conditions by the function boundary_conditions.cc.
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VII. |
Acknowledgements
The authors would like to thank Chris Anderson (UCLA Mathematics) for contributing the original source code for the class of multidimensional arrays used in this package, and Nick Kisseberth (former CSCAMM Computer System Administrator) for his help in organazing and simplifying the source code.
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