Numerical methods for partial differential equations

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

Scientific Background
Motivation of this area
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 methodHirota, R. (2004). The direct method in soliton theory (Vol. 155). Cambridge University Press. and the inverse scattering method.Novikov, S., Manakov, S. V., Pitaevskii, L. P., & Zakharov, V. E. (1984). Theory of solitons: the inverse scattering method. Springer Science & Business Media.Ablowitz, M. J., & Segur, H. (1981). Solitons and the inverse scattering transform (Vol. 4). SIAM. This is why numerical methods for PDEs are needed.
The Finite Difference Method (FDM) and its problems
One of the most basic PDE solver is the finite difference method (FDM).Strikwerda, J. C. (2004). Finite difference schemes and partial differential equations. SIAM. This method approximates derivatives as differences:

This method works for easy problems. But it is powerless to some equations (such as the Navier–Stokes equationsConstantin, P., & Foias, C. (1988). Navier-stokes equations. University of Chicago Press.Temam, R. (2001). Navier-Stokes equations: theory and numerical analysis (Vol. 343). American Mathematical Society.Foias, C., Manley, O., Rosa, R., & Temam, R. (2001). Navier-Stokes equations and turbulence (Vol. 83). Cambridge University Press.Girault, V., & Raviart, P. A. (2012). Finite element methods for Navier-Stokes equations: theory and algorithms (Vol. 5). Springer Science & Business Media.) because they are non-linear. Since this difficulty appeared, numerical analysts started to study other methods (just like the finite element method,Brenner, S., & Scott, R. (2007). The mathematical theory of finite element methods (Vol. 15). Springer Science & Business Media.Oden, J. T., & Reddy, J. N. (2012). An introduction to the mathematical theory of finite elements. Courier Corporation. FEM). On the other hand, some experts started to consider improvements for FDM.

Evolution of the FDM
Experts have discovered difference methods which preserves the property of the given PDE.
Integrable Difference Schemes
Ryogo Hirota,R. Hirota, J. Phys. Soc. Jpn. 43 (1977) 4116-4124.R. Hirota, J. Phys. Soc. Jpn. 43 (1977) 2074-2078.R. Hirota, J. Phys. Soc. Jpn. 43 (1977) 2079-2086.R. Hirota, J. Phys. Soc. Jpn. 45 (1978) 321-332.R. Hirota, J. Phys. Soc. Jpn. 46 (1979) 312-319. Mark Ablowitz and othersM. J. Ablowitz and J. F. Ladik, J. Math. Phys. 16 (1975) 598-603.M. J. Ablowitz and J. F. Ladik, J. Math. Phys. 17 (1976) 1011-1018.M. J. Ablowitz and J. F. Ladik, Stud. Appl. Math. 55 (1977) 213-229.M. J. Ablowitz and J. F. Ladik, Stud. Appl. Math. 57 (1977) 1-12. 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.T. R. Taha and M. J. Ablowitz, J. Comput. Phys., 55 (1984), 192-202.T. R. Taha and M. J. Ablowitz, J. Comput. Phys., 55 (1984), 203-230.T. R. Taha and M. J. Ablowitz, J. Comput. Phys., 55 (1984), 231-253.T. R. Taha and M. J. Ablowitz, J. Comput. Phys., 55 (1988), 540-548.
Structure Preserving Numerical Methods
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:
 Symplectic integratorsHairer, E., Lubich, C., & Wanner, G. (2006). Geometric numerical integration: structure-preserving algorithms for ordinary differential equations. Springer Science & Business Media.
 Discrete gradient methodGonzalez, O., Time integration and discrete Hamiltonian systems, J. Nonlinear Sci., 6 (1996), 449‐467.Greenspan, D., Discrete Models, in Applied Mathematics and Computation, Addison‐Wesley Publishing Co., Reading, 1973.A. Ishikawa and T. Yaguchi, Geometric investigation of the discrete gradient method for the Webster equation with a weighted inner product, JSIAM Lett., 7 (2015), 17-20.A. Ishikawa and T. Yaguchi, Invariance of Furihata's Discrete Gradient Schemes for the Webster Equation with Different Riemannian Structures, AIP Conf. Proc. 1648, 180003 (2015).
 Discrete variational derivative method (DVDM)Furihata, D., & Matsuo, T. (2010). Discrete variational derivative method: a structure-preserving numerical method for partial differential equations. Chapman and Hall/CRC.
Some experts are studying their relation between numerical linear algebra.Miyatake, Y., Sogabe, T., & Zhang, S. L. (2018). On the equivalence between SOR-type methods for linear systems and the discrete gradient methods for gradient systems. Journal of Computational and Applied Mathematics, 342, 58-69.
Others
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:
 Shortley-Weller approximationBramble, J. H., & Hubbard, B. E. (1962). On the formulation of finite difference analogues of the Dirichlet problem for Poisson's equation. Numerische Mathematik, 4(1), 313-327.Matsunaga, N., & Yamamoto, T. (2000). Superconvergence of the Shortley–Weller approximation for Dirichlet problems. Journal of Computational and Applied Mathematics, 116(2), 263-273.Yamamoto T. (2001) A New Insight of the Shortley-Weller Approximation for Dirichlet Problems. In: Alefeld G., Rohn J., Rump S., Yamamoto T. (eds) Symbolic Algebraic Methods and Verification Methods. Springer, Vienna
 Swarztrauber-Sweet approximationSwarztrauber, P. N., & Sweet, R. A. (1973). The direct solution of the discrete Poisson equation on a disk. SIAM Journal on Numerical Analysis, 10(5), 900-907.Strikwerda, J. C., & Nagel, Y. M. (1986). A Numerical Method for the Incompressible Navier-Stokes Equations in Three-Dimensional Cylindrical Geometry (No. MRC-TSR-2948). WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER.Matsunaga, N., & Yamamoto, T. (1999). Convergence of swartztrauber-sweets approximation for the poisson-type equation on a disk. Numerical functional analysis and optimization, 20(9-10), 917-928.
 Ascher-Mattheij-Russell difference formulaFang, Q. (2006). Convergence of Ascher-Mattheij-Russell Finite Difference Method for a Class of Two-point Boundary Value Problems. INFORMATION, 9(4), 563.Zhang, X. Y. (2010). A New Ascher-Mattheij-Russell Type FDM for Nonlinear Two-point Boundary Value Problems. INFORMATION-AN INTERNATIONAL INTERDISCIPLINARY JOURNAL, 13(4), 1185-1194.
Validated Numerics for PDEs

Not only approximate solvers, but the study to "verify the existence of solution by computers" is also active.M. Nakao, M. Plum, Y. Watanabe (2019) Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations (Springer Series in Computational Mathematics). This study is needed because numerically obtained solutions could be phantom solutions (fake solutions). This kind of incident is already reported.Breuer, B., Plum, M., & McKenna, P. J. (2001). "Inclusions and existence proofs for solutions of a nonlinear boundary value problem by spectral numerical methods." In Topics in Numerical Analysis (pp. 61–77). Springer, Vienna. Gidas, B., Ni, W. M., & Nirenberg, L. (1979). "Symmetry and related properties via the maximum principle." Communications in Mathematical Physics, 68(3), 209–243.
PDEs that have been Studied in the Context of Validated Numerics

Journal
The scientific journal "Numerical Methods for Partial Differential Equations" is published to promote the studies of this area.Numerical Methods for Partial Differential Equations, Wiley Online Library
Related Software
Chebfun is one of the most famous software in this field.Driscoll, T. A., Hale, N., & Trefethen, L. N. (2014). Chebfun guide.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.Hashemi, B., & Trefethen, L. N. (2017). Chebfun in three dimensions. SIAM Journal on Scientific Computing, 39(5), C341-C363.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. They are also many libraries based on the finite element method such as:
 FreeFem++Hecht, F. (2012). New development in FreeFem++. Journal of numerical mathematics, 20(3-4), 251-266.Hecht, F., Pironneau, O., Le Hyaric, A., & Ohtsuka, K. (2005). FreeFem++ manual.Sadaka, G. (2012). FreeFem++, a tool to solve PDEs numerically. arXiv preprint arXiv:1205.1293.
 FEniCS ProjectAlnæs, 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).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.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.Langtangen, H. P., Logg, A., & Tveito, A. (2016). Solving PDEs in Python: The FEniCS Tutorial I. Springer International Publishing.

Experts in this Field

 Israel Gelfand
 John von Neumann
 Masatake Mori
 Peter Deuflhard
 Peter Lax
 Shinichi Oishi

References

Further Reading
 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 Method

 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 Method
 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 Method
 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 Method

 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 Method

 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 Methods

 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.

Category:Partial differential equations
Category:Numerical analysis