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Difference Methods for PDE in Several Dimensions

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Book cover Computational Methods for Physicists

Part of the book series: Graduate Texts in Physics ((GTP))

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Abstract

Finite-difference methods for partial differential equations in several dimensions are presented by first handling the basic (parabolic) diffusion equation in two dimensions by explicit and implicit difference schemes, allowing us to introduce the corresponding stability criteria. The treatment of alternating direction implicit (ADI) schemes is supplemented by the extension to three space dimensions. Several schemes for solutions of hyperbolic equations are listed. Solving elliptic equations (Poisson and Laplace) by classical relaxation methods (Jacobi, Gauss–Seidel, SOR, SSOR) is discussed, as well as high-resolution schemes for hyperbolic equations. The need to respect the underlying nature of the physical problems is emphasized by solving the two-dimensional diffusion and Poisson equations in polar coordinates with various boundary conditions. The boundary element method, the finite element method, and a mesh-free method based on radial basis functions are presented. In addition to many variants of Laplace and Poisson equations, the Problems include several instances of the linear diffusion equation, as well as a non-linear case describing the growth of bacterial biofilms.

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References

  1. J.W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods. Springer Texts in Applied Mathematics, vol. 22 (Springer, Berlin, 1998)

    Google Scholar 

  2. J.W. Thomas, Numerical Partial Differential Equations: Conservation Laws and Elliptic Equations. Springer Texts in Applied Mathematics, vol. 33 (Springer, Berlin, 1999)

    MATH  Google Scholar 

  3. W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes: The Art of Scientific Computing, 3rd edn. (Cambridge University Press, Cambridge, 2007). See also the equivalent handbooks in Fortran, Pascal and C, as well as http://www.nr.com

    MATH  Google Scholar 

  4. R.A. Horn, C.R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1985)

    MATH  Google Scholar 

  5. J.W. Demmel, Applied Numerical Linear Algebra (SIAM, Philadelphia, 1997)

    Book  MATH  Google Scholar 

  6. J.R. Shewchuk, An Introduction to the Conjugate Gradient Method Without the Agonizing Pain (Carnegie Mellon University, Pittsburgh, 1994) (unpublished, but accessible through numerous websites)

    Google Scholar 

  7. G.H. Golub, C.F. Van Loan, Matrix Computations, 3rd edn. (Johns Hopkins University Press, Baltimore, 1996)

    MATH  Google Scholar 

  8. S.T. Zalesak, Fully multidimensional flux-corrected transport algorithms for fluids. J. Comput. Phys. 31, 335 (1979)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. P.K. Smolarkiewicz, A fully multidimensional positive definite advection transport algorithm with small implicit diffusion. J. Comput. Phys. 54, 325 (1984)

    Article  ADS  Google Scholar 

  10. N.A. Peterson, An algorithm for assembling overlapping grid systems. SIAM J. Sci. Comput. 20, 1995 (1999)

    Article  MathSciNet  Google Scholar 

  11. W.D. Henshaw, On multigrid for overlapping grids. SIAM J. Sci. Comput. 26, 1547 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. W.T. Ang, A Beginner’s Course in Boundary Element Methods (Universal, Boca Raton, 2007)

    Google Scholar 

  13. J.E. Flaherty, Finite Element Analysis. CSCI, MATH Lecture Notes, vol. 6860 (Rensselaer Polytechnic Institute, Troy, 2000)

    Google Scholar 

  14. Z. Chen, Finite Element Methods and Their Applications (Springer, Berlin, 2005)

    MATH  Google Scholar 

  15. M.S. Gockenbach, Understanding and Implementing the Finite Element Method (SIAM, Philadelphia, 2006)

    Book  MATH  Google Scholar 

  16. Computational Geometry Algorithms Library. http://www.cgal.org. The algorithms from this library are also built into Matlab

  17. M. de Berg, O. Cheong, M. van Kreveld, M. Overmars, Computational Geometry: Algorithms and Applications, 3rd edn. (Springer, Berlin, 2008)

    MATH  Google Scholar 

  18. J.R. Shewchuk, Delaunay refinement algorithms for triangular mesh generation. Comput. Geom. 22, 21 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. J. Alberty, C. Carstensen, S.A. Funken, Remarks around 50 lines of Matlab: short finite element implementation. Numer. Algorithms 20, 117 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. J. Alberty, C. Carstensen, S.A. Funken, R. Klose, Matlab implementation of the finite element method in elasticity. Computing 69, 239 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. http://en.wikipedia.org/wiki/List_of_finite_element_software_packages

  22. F. Hecht, O. Pironneau, J. Morice, A. Le Hyaric, K. Ohtsuka, FreeFem++. http://www.freefem.org/ff++

  23. D. Knoll, J. Morel, L. Margolin, M. Shashkov, Physically motivated discretization methods. Los Alamos Sci. 29, 188 (2005)

    Google Scholar 

  24. M. Shashkov, S. Steinberg, Solving diffusion equations with rough coefficients on rough grids. J. Comput. Phys. 129, 383 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. J. Hyman, M. Shashkov, S. Steinberg, The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials. J. Comput. Phys. 132, 130 (1997)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. J. Hyman, M. Shashkov, Mimetic discretizations for Maxwell’s equations. J. Comput. Phys. 151, 881 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. J. Hyman, J. Morel, M. Shashkov, S. Steinberg, Mimetic finite difference methods for diffusion equations. Comput. Geosci. 6, 333 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Y. Kuznetsov, K. Lipnikov, M. Shashkov, The mimetic finite difference method on polygonal meshes for diffusion-type problems. Comput. Geosci. 8, 301 (2004)

    Article  MathSciNet  Google Scholar 

  29. P. Bochev, J. Hyman, Principles of mimetic discretizations of differential operators, in Compatible Spatial Discretizations, ed. by D.N. Arnold, P.B. Bochev, R.B. Lehoucq, R.A. Nicolaides, M. Shashkov. The IMA Volumes in Mathematics and Its Applications, vol. 142 (Springer, Berlin, 2006), p. 89

    Chapter  Google Scholar 

  30. Multiple authors, Special issue of Comput. Sci. Eng. Nov/Dec (2006)

    Google Scholar 

  31. P. Wesseling, An Introduction to Multigrid Methods (Edwards, Philadelphia, 2004)

    Google Scholar 

  32. W.L. Briggs, H. van Emden, S.F. McCormick, A Multigrid Tutorial, 2nd edn. (SIAM, Philadelphia, 2000)

    Book  MATH  Google Scholar 

  33. S. Li, W.K. Liu, Mesh-Free Particle Methods (Springer, Berlin, 2004)

    Google Scholar 

  34. G.R. Liu, Mesh-Free Methods: Moving Beyond the Finite-Element Method (CRC Press, Boca Raton, 2003)

    Book  MATH  Google Scholar 

  35. E.J. Kansa, Multiquadrics—a scattered data approximation scheme with applications to computational fluid dynamics, I: surface approximations and partial derivative estimates. Comput. Math. Appl. 19, 127 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  36. E.J. Kansa, Multiquadrics—a scattered data approximation scheme with applications to computational fluid dynamics, II: solutions of parabolic, hyperbolic and elliptic partial differential equations. Comput. Math. Appl. 19, 147 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  37. N. Flyer, B. Fornberg, Radial basis functions: developments and applications to planetary scale flows. Comput. Fluids 46, 23 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  38. M. Tatari, M. Dehghan, A method for solving partial differential equations via radial basis functions: application to the heat equation. Eng. Anal. Bound. Elem. 34, 206 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  39. E.J. Kansa, Exact explicit time integration of hyperbolic partial differential equations with mesh free radial basis functions. Eng. Anal. Bound. Elem. 31, 577 (2007)

    Article  MATH  Google Scholar 

  40. E. Larsson, B. Fornberg, A numerical study of some radial basis function based solution methods for elliptic PDEs. Comput. Math. Appl. 46, 891 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  41. C.-S. Huang, H.-D. Yen, A.H.-D. Cheng, On the increasingly flat radial basis function and optimal shape parameter for the solution of elliptic PDEs. Eng. Anal. Bound. Elem. 34, 802 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  42. H.J. Eberl, L. Demaret, A finite difference scheme for a degenerated diffusion equation arising in microbial biology. Electron. J. Differ. Equ. 15, 77 (2007)

    MathSciNet  Google Scholar 

  43. I. Klapper, J. Dockery, Mathematical description of microbial biofilms. SIAM Rev. 52, 221 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia

    Simon Širca & Martin Horvat

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  1. Simon Širca
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  2. Martin Horvat
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Širca, S., Horvat, M. (2012). Difference Methods for PDE in Several Dimensions. In: Computational Methods for Physicists. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32478-9_10

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