Solvers of Conservation Laws

SOLVCON is a collection of Python-based conservation-law solvers that use the space-time Conservation Element and Solution Element (CESE) method [Chang95]. SOLVCON targets at solving problems that can be formulated as a system of first-order, linear or non-linear partial differential equations (PDEs) [Lax73]:

\dpd{\bvec{u}}{t}
+ \sum_{k=1}^3 \mathrm{A}^{(k)}(\bvec{u})\dpd{\bvec{u}}{x_k}
= \bvec{s}(\bvec{u})

where \bvec{u} is the unknown vector, \mathrm{A}^{(1)}, \mathrm{A}^{(2)}, and \mathrm{A}^{(3)} the Jacobian matrices, and \bvec{s} the source term. SOLVCON is designed to be a software framework to house various solvers. The design of SOLVCON is discussed in [Chen11].

Get the Code and the Dependencies

Please clone the development version from BitBucket (using Mercurial):

hg clone https://bitbucket.org/solvcon/solvcon

SOLVCON has the following dependencies: gcc 4.3+, SCons 2+, Python 2.7, Cython 0.16+, Numpy 1.5+, LAPACK, NetCDF 4+, SCOTCH 6.0+, Nose 1.0+, Paramiko 1.14+, boto 2.29.1+, gmsh 2.5+, and VTK 5.6+. You can install them by running the scripts aptget.*.sh (Debian/Ubuntu) or conda.sh (Miniconda/Anaconda) provided in the contrib/ directory.

Build

The binary part of SOLVCON should be built with SCons:

scons scmods

After worth, it can be built with distutils:

python setup.py build_ext --inplace

SOLVCON needs not explicit installation. Setting the environment variables $PATH and $PYTHONPATH is sufficient.

Building document requires Sphinx 1.1.2+, Sphinxcontrib issue tracker 0.11, and graphviz 2.28+. Once the binary of SOLVCON is built, the following commands can build the document:

cd doc
make html

The built document will be available at doc/build/html/.

Run Tests

Tests should be run with Nose:

nosetests

Another set of tests are collected in ftests/ directory, and can be run with:

nosetests ftests/*

Some tests in ftests/ involve remote procedure call (RPC) that uses ssh. You need to set up the public key authentication to properly run them.

Appendices

Other Resources

  • Papers and presentations:
  • The CESE method:
    • The CE/SE working group: http://www.grc.nasa.gov/WWW/microbus/
    • The CESE research group at OSU: http://cfd.solvcon.net/research.html
    • Selected papers:
      • Sin-Chung Chang, “The Method of Space-Time Conservation Element and Solution Element – A New Approach for Solving the Navier-Stokes and Euler Equations”, Journal of Computational Physics, Volume 119, Issue 2, July 1995, Pages 295-324. doi: 10.1006/jcph.1995.1137
      • Xiao-Yen Wang, Sin-Chung Chang, “A 2D Non-Splitting Unstructured Triangular Mesh Euler Solver Based on the Space-Time Conservation Element and Solution Element Method”, Computational Fluid Dynamics Journal, Volume 8, Issue 2, 1999, Pages 309-325.
      • Zeng-Chan Zhang, S. T. John Yu, Sin-Chung Chang, “A Space-Time Conservation Element and Solution Element Method for Solving the Two- and Three-Dimensional Unsteady Euler Equations Using Quadrilateral and Hexahedral Meshes”, Journal of Computational Physics, Volume 175, Issue 1, Jan. 2002, Pages 168-199. doi: 10.1006/jcph.2001.6934
  • Related Links
  • Other Software for Solving PDEs

Contributors

Bibliography

[Anderson03]John David Anderson, Modern Compressible Flow: With Historical Perspective, McGraw-Hill, 2003. ISBN 0072424435.
[Chang95]Sin-Chung Chang, “The Method of Space-Time Conservation Element and Solution Element – A New Approach for Solving the Navier-Stokes and Euler Equations”, Journal of Computational Physics, Volume 119, Issue 2, July 1995, Pages 295-324. doi: 10.1006/jcph.1995.1137
[Chen11]Yung-Yu Chen, A Multi-Physics Software Framework on Hybrid Parallel Computing for High-Fidelity Solutions of Conservation Laws, Ph.D. Thesis, The Ohio State University, United States, Aug. 2011. (OhioLINK)
[Chen12]Yung-Yu Chen, Lixiang Yang, and Sheng-Tao John Yu, “Hyperbolicity of Velocity-Stress Equations for Waves in Anisotropic Elastic Solids”, Journal of Elasticity, Volume 106, Issue 2, Feb. 2012, Page 149-164. doi: 10.1007/s10659-011-9315-8
[Lax73]Peter D. Lax, “Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves”, Society for Industrial Mathematics, 1973. ISBN 0898711770.
[Mavriplis97]D. J. Mavriplis, Unstructured grid techniques, Annual Review of Fluid Mechanics 29. (1997)
[Warming75]R. F. Warming, Richard M. Beam, and B. J. Hyett, “Diagonalization and Simultaneous Symmetrization of the Gas-Dynamic Matrices”, Mathematics of Computation, Volume 29, Issue 132, Oct. 1975, Page 1037-1045. http://www.jstor.org/stable/2005742
[Yang13]Lixiang Yang, Yung-Yu Chen, Sheng-Tao John Yu, “Viscoelasticity determined by measured wave absorption coefficient for modeling waves in soft tissues”, Wave Motion, Volume 50, Issue 2, March 2013, Page 334-346. doi: 10.1016/j.wavemoti.2012.09.002.

Indices and Tables