An implicit finite volume method for arbitrary transport equations

Dalton James Eric Harvie

Abstract


A finite volume framework is described for solving multiphysics transport problems. The method operates in a unique way. The transport equations and associated boundary conditions are input by the user using pseudo-mathematical expressions. A Perl program parses these equations and, via the computer algebra system Maxima, `metaprograms' a Fortran code that solves the problem on an unstructured mesh using the Newton--Raphson method. The strength of the technique is that a fully implicit numerical formulation is generated and modified easily, for an arbitrary set of equations. The implemented algorithm (`arb') is available for download and licensed under the GNU General Public License.

References
  • Luis Cueto-Felgueroso, Ignasi Colominas, Xesus Nogueira, Fermin Navarrina, and Manuel Casteleiro. Finite volume solvers and moving least-squares approximations for the compressible Navier--Stokes equations on unstructured grids. Computer Methods in Applied Mechanics and Engineering, 196(45--48):4712--4736, 2007. doi:10.1016/j.cma.2007.06.003.
  • Timothy A. Davis. {Algorithm 832: UMFPACK V4.3---an unsymmetric-pattern multifrontal method}. ACM Trans. Math. Softw., 30(2):196--199, June 2004. ISSN 0098-3500. doi:10.1145/992200.992206. http://www.cise.ufl.edu/research/sparse/umfpack/.
  • J. H. Ferziger and M. Peric. Computational Methods for Fluid Dynamics. Springer-Verlag, 3rd edition, 2002.
  • CSC IT Center for Science. Elmer: Open source finite element software for multiphysical problems. http://www.csc.fi/english/pages/elmer/. Accessed 31/1/11.
  • Christophe Geuzaine and Jean-Francois Remacle. Gmsh: A 3-d finite element mesh generator with built-in pre- and post-processing facilities. International Journal for Numerical Methods in Engineering, 79(11):1309--1331, 2009. ISSN 1097-0207. doi:10.1002/nme.2579.
  • The COMSOL Group. Comsol multiphysics. http://www.comsol.com/products/multiphysics/. Accessed 31/1/11.
  • Dalton Harvie. arb manual: version 0.25, 2011. http://www.chemeng.unimelb.edu.au/people/staff/daltonh/downloads/arb/code/manual.pdf.
  • J. E. Dennis {Jr.} and Robert B. Schnabel. Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, 1983. ISBN 0-13-627216-9.
  • William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery. Numerical Recipes in FORTRAN: The art of scientific computing. Cambridge University Press, Second edition, 1992.
  • William Schelter, Maxima Users, and Developers Group. Maxima: A computer algebra system, 2011. http://maxima.sourceforge.net/.
  • Olaf Schenk and Klaus Gartner. Solving unsymmetric sparse systems of linear equations with pardiso. Future Generation Computer Systems, 20(3):475--487, 2004. ISSN 0167-739X. doi:10.1016/j.future.2003.07.011. http://www.pardiso-project.org/. Selected numerical algorithms.

Keywords


finite volume; transport equations

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DOI: http://dx.doi.org/10.21914/anziamj.v52i0.3949



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