Numerical methods for partial differential equations

class of methods for solving partial differential equations

Numerical methods for partial differential equations are computational schemes to obtain approximate solutions of partial differential equations (PDEs).

JournalEdit

The scientific journal "Numerical Methods for Partial Differential Equations" is published to promote the studies of this area[1].

Related SoftwareEdit

Chebfun is one of the most famous software in this field[2][3][4][5]. They are also many libraries based on the finite element method such as:

Scientific BackgroundEdit

Motivation of this areaEdit

Many PDEs appeared for the study of physics and other areas in science. Therefore, many mathematicians have challenged to make methods to solve them, but there is no method to mathematically solve PDEs except the Hirota direct method[13] and the inverse scattering method[14][15]. This is why numerical methods for PDEs are needed.

The Finite Difference Method (FDM) and its problemsEdit

One of the most basic PDE solver is the finite difference method (FDM)[16]. This method approximates derivatives as differences:

 

This method works for easy problems. But it is powerless to some equations (such as the Navier–Stokes equations[17][18][19][20]) because they are non-linear. Since this difficulty appeared, numerical analysts started to study other methods (just like the finite element method[21][22], FEM). On the other hand, some experts started to consider improvements for FDM.

Evolution of the FDMEdit

Experts have discovered difference methods which preserves the property of the given PDE.

Integrable Difference SchemesEdit

Ryogo Hirota[23][24][25][26][27], Mark Ablowitz and others[28][29][30][31] have made methods that preserves the integrability (important mathematical property in the theory of dynamical systems) of PDEs. These methods are known to have better accuaracy than the original FDM[32][33][34][35].

Structure Preserving Numerical MethodsEdit

Many PDEs have appeared from physics. So we can think about difference methods preserving physical properties. These difference methods are known as structure preserving numerical methods. The following list is the examples of them:

Some experts are studying their relation between numerical linear algebra[42].

OthersEdit

The difference mthods in above have high accuracy, but their usage is limited because they depend on the behaviour of the given PDEs. This is why new types of FDM are still studied. For example, the following methods are studied:

Validated Numerics for PDEsEdit

Not only approximate solvers, but the study to "verify the existence of solution by computers" is also active[51]. This study is needed because numerically obtained solutions could be phantom solutions (fake solutions). This kind of incident is already reported[52][53].

PDEs that have been Studied in the Context of Validated NumericsEdit

Experts in this FieldEdit

ReferencesEdit

  1. Numerical Methods for Partial Differential Equations, Wiley Online Library
  2. Driscoll, T. A., Hale, N., & Trefethen, L. N. (2014). Chebfun guide.
  3. Platte, R. B., & Trefethen, L. N. (2010). Chebfun: a new kind of numerical computing. In Progress in industrial mathematics at ECMI 2008 (pp. 69-87). Springer, Berlin, Heidelberg.
  4. Hashemi, B., & Trefethen, L. N. (2017). Chebfun in three dimensions. SIAM Journal on Scientific Computing, 39(5), C341-C363.
  5. Wright, G. B., Javed, M., Montanelli, H., & Trefethen, L. N. (2015). Extension of Chebfun to periodic functions. SIAM Journal on Scientific Computing, 37(5), C554-C573.
  6. Hecht, F. (2012). New development in FreeFem++. Journal of numerical mathematics, 20(3-4), 251-266.
  7. Hecht, F., Pironneau, O., Le Hyaric, A., & Ohtsuka, K. (2005). FreeFem++ manual.
  8. Sadaka, G. (2012). FreeFem++, a tool to solve PDEs numerically. arXiv preprint arXiv:1205.1293.
  9. Alnæs, M., Blechta, J., Hake, J., Johansson, A., Kehlet, B., Logg, A., ... & Wells, G. N. (2015). The FEniCS project version 1.5. Archive of Numerical Software, 3(100).
  10. Dupont, T., Hoffman, J., Johnson, C., Kirby, R. C., Larson, M. G., Logg, A., & Scott, L. R. (2003). The fenics project. Chalmers Finite Element Centre, Chalmers University of Technology.
  11. Logg, A., Mardal, K. A., & Wells, G. (Eds.). (2012). Automated solution of differential equations by the finite element method: The FEniCS book. en:Springer Science & Business Media.
  12. Langtangen, H. P., Logg, A., & Tveito, A. (2016). Solving PDEs in Python: The FEniCS Tutorial I. Springer International Publishing.
  13. Hirota, R. (2004). The direct method in soliton theory (Vol. 155). Cambridge University Press.
  14. Novikov, S., Manakov, S. V., Pitaevskii, L. P., & Zakharov, V. E. (1984). Theory of solitons: the inverse scattering method. Springer Science & Business Media.
  15. Ablowitz, M. J., & Segur, H. (1981). Solitons and the inverse scattering transform (Vol. 4). SIAM.
  16. Strikwerda, J. C. (2004). Finite difference schemes and partial differential equations. SIAM.
  17. Constantin, P., & Foias, C. (1988). Navier-stokes equations. University of Chicago Press.
  18. Temam, R. (2001). Navier-Stokes equations: theory and numerical analysis (Vol. 343). American Mathematical Society.
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  41. Furihata, D., & Matsuo, T. (2010). Discrete variational derivative method: a structure-preserving numerical method for partial differential equations. Chapman and Hall/CRC.
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  61. Zgliczynski, P. (2002). Attracting fixed points for the Kuramoto--Sivashinsky equation: A computer assisted proof. SIAM Journal on Applied Dynamical Systems, 1(2), 215-235.
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  63. Zgliczynski, P., & Mischaikow, K. (2001). Rigorous numerics for partial differential equations: The Kuramoto—Sivashinsky equation. Foundations of Computational Mathematics, 1(3), 255-288.
  64. Watanabe, Y., Yamamoto, N., & Nakao, M. T. (1999). A numerical verification method of solutions for the Navier-Stokes equations. In Developments in reliable computing (pp. 347-357). Springer, Dordrecht.
  65. Nakao, M. T., Hashimoto, K., & Kobayashi, K. (2007). Verified numerical computation of solutions for the stationary Navier-Stokes equation in nonconvex polygonal domains. Hokkaido Mathematical Journal, 36(4), 777-799.
  66. Lahmann, J. R., & Plum, M. (2004). A computer‐assisted instability proof for the Orr‐Sommerfeld equation with Blasius profile. ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik: Applied Mathematics and Mechanics, 84(3), 188-204.
  67. Watanabe, Y., Plum, M., & Nakao, M. T. (2009). A computer‐assisted instability proof for the Orr‐Sommerfeld problem with Poiseuille flow. ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik: Applied Mathematics and Mechanics, 89(1), 5-18.
  68. Watanabe, Y., Nagatou, K., Plum, M., & Nakao, M. T. (2011). A computer-assisted stability proof for the Orr-Sommerfeld problem with Poiseuille flow. Nonlinear Theory and Its Applications, IEICE, 2(1), 123-127.

Further ReadingEdit

  • Iserles, A. (2009). A first course in the numerical analysis of differential equations. Cambridge University Press.
  • Computational Partial Differential Equations Using MATLAB, Jichun Li and Yi-Tung Chen, Chapman & Hall.
  • Ames, W. F. (2014). Numerical methods for partial differential equations. Academic Press.
  • Ganzha, V. G. E., & Vorozhtsov, E. V. (1996). Numerical solutions for partial differential equations: problem solving using Mathematica. CRC Press.

Literatures for specific solvers are described as follows.

Finite Element MethodEdit

  • Brenner, S., & Scott, R. (2007). The mathematical theory of finite element methods. Springer Science & Business Media.
  • Johnson, C. (2012). Numerical solution of partial differential equations by the finite element method. Courier Corporation.
  • Strang, G., & Fix, G. J. (1973). An analysis of the finite element method. Englewood Cliffs, NJ: Prentice-hall.
  • Boffi, D., Brezzi, F., & Fortin, M. (2013). Mixed finite element methods and applications. Heidelberg: Springer.
  • Braess, D. (2007). Finite elements: Theory, fast solvers, and applications in solid mechanics. Cambridge University Press.

Finite Difference MethodEdit

  • Smith, G. D. (1985). Numerical solution of partial differential equations: finite difference methods. Oxford University Press.
  • Strikwerda, J. C. (2004). Finite difference schemes and partial differential equations. SIAM.

Finite Volume MethodEdit

  • Eymard, R. Gallouët, T. R. Herbin, R. (2000) The finite volume method, in Handbook of Numerical Analysis, Vol. VII, 2000, p. 713–1020. Editors: P.G. Ciarlet and J.L. Lions.
  • LeVeque, Randall (2002), Finite Volume Methods for Hyperbolic Problems, Cambridge University Press.

Boundary Element MethodEdit

  • Banerjee, Prasanta Kumar (1994), The Boundary Element Methods in Engineering (2nd ed.), London, etc.: McGraw-Hill.
  • Beer, Gernot; Smith, Ian; Duenser, Christian, The Boundary Element Method with Programming: For Engineers and Scientists, Berlin – Heidelberg – New York: Springer-Verlag, pp. XIV+494.
  • Cheng, Alexander H.-D.; Cheng, Daisy T. (2005), "Heritage and early history of the boundary element method", Engineering Analysis with Boundary Elements, 29 (3): 268–302.
  • Katsikadelis, John T. (2002), Boundary Elements Theory and Applications, Amsterdam: Elsevier, pp. XIV+336.
  • Wrobel, L. C.; Aliabadi, M. H. (2002), The Boundary Element Method, New York: John Wiley & Sons, p. 1066, (in two volumes).

Spectral MethodEdit

  • Lloyd N. Trefethen (2000) Spectral Methods in MATLAB. SIAM, Philadelphia, PA.
  • D. Gottlieb and S. Orzag (1977) "Numerical Analysis of Spectral Methods : Theory and Applications", SIAM, Philadelphia, PA.
  • J. Hesthaven, S. Gottlieb and D. Gottlieb (2007) "Spectral methods for time-dependent problems", Cambridge UP, Cambridge, UK.
  • Canuto C., Hussaini M. Y., Quarteroni A., and Zang T.A. (2006) Spectral Methods. Fundamentals in Single Domains. Springer-Verlag, Berlin Heidelberg

Structure Preserving Numerical MethodsEdit

  • Leimkuhler, B. and Reich, S., Simulating Hamiltonian Dynamics, Cambridge University Press, Cambridge, 2004.
  • Sanz‐Serna, J. M. and Calvo, M. P., Numerical Hamiltonian Problems, Applied Mathematics and Mathematical Computation 7, Chapman & Hall, London, 1994.
  • Arnold, D. N., Bochev, P. B., Lehoucq, R. B., Nicolaides, R. A. and Shashkov, M. (eds.), Compatible Spatial Discretizations, in The IMA Volumes in Mathematics and Its Applications, Springer, New York, 2006.
  • Budd, C. and Piggott, M. D., Geometric integration and its applications, in Handbook of Numerical Analysis, XI, North‐Holland, Amsterdam, 2003, 35‐139.
  • Christiansen, S. H., Munthe‐Kaas, H. Z. and Owren, B., Topics in structure‐preserving discretization, Acta Numerica, 20 (2011), 1‐119.
  • Shashkov, M., Conservative Finite‐Difference Methods on General Grids, CRC Press, Boca Raton, 1996.