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 method and the inverse scattering method. 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). 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) because they are non-linear. Since this difficulty appeared, numerical analysts started to study other methods (just like the finite element method, 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, Mark Ablowitz and others 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.
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 integrators
Discrete gradient method
Discrete variational derivative method (DVDM)
Some experts are studying their relation between numerical linear algebra.
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 approximation
Swarztrauber-Sweet approximation
Ascher-Mattheij-Russell difference formula
Validated Numerics for PDEs
Not only approximate solvers, but the study to "verify the existence of solution by computers" is also active. This study is needed because numerically obtained solutions could be phantom solutions (fake solutions). This kind of incident is already reported.
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.
Related Software
Chebfun is one of the most famous software in this field. They are also many libraries based on the finite element method such as:
FreeFem++
FEniCS Project
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. Gallouet, 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.
Partial differential equations
Numerical analysis