# 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[1] and the inverse scattering method.[2][3] 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).[4] This method approximates derivatives as differences:

${\displaystyle f^{\prime }(x)\simeq {\frac {f(x+h)-f(x)}{h}},\quad h\ll 1.}$

This method works for easy problems. But it is powerless to some equations (such as the Navier–Stokes equations[5][6][7][8]) because they are non-linear. Since this difficulty appeared, numerical analysts started to study other methods (just like the finite element method,[9][10] 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,[11][12][13][14][15] Mark Ablowitz and others[16][17][18][19] 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.[20][21][22][23]

#### 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:

Some experts are studying their relation between numerical linear algebra.[30]

#### 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:

## Validated Numerics for PDEs

Not only approximate solvers, but the study to "verify the existence of solution by computers" is also active.[39] This study is needed because numerically obtained solutions could be phantom solutions (fake solutions). This kind of incident is already reported.[40][41]

## Journal

The scientific journal "Numerical Methods for Partial Differential Equations" is published to promote the studies of this area.[57]

## Related Software

Chebfun is one of the most famous software in this field.[58][59][60][61] They are also many libraries based on the finite element method such as:

## References

1. Hirota, R. (2004). The direct method in soliton theory (Vol. 155). Cambridge University Press.
2. Novikov, S., Manakov, S. V., Pitaevskii, L. P., & Zakharov, V. E. (1984). Theory of solitons: the inverse scattering method. Springer Science & Business Media.
3. Ablowitz, M. J., & Segur, H. (1981). Solitons and the inverse scattering transform (Vol. 4). SIAM.
4. Strikwerda, J. C. (2004). Finite difference schemes and partial differential equations. SIAM.
5. Constantin, P., & Foias, C. (1988). Navier-stokes equations. University of Chicago Press.
6. Temam, R. (2001). Navier-Stokes equations: theory and numerical analysis (Vol. 343). American Mathematical Society.
7. Foias, C., Manley, O., Rosa, R., & Temam, R. (2001). Navier-Stokes equations and turbulence (Vol. 83). Cambridge University Press.
8. Girault, V., & Raviart, P. A. (2012). Finite element methods for Navier-Stokes equations: theory and algorithms (Vol. 5). Springer Science & Business Media.
9. Brenner, S., & Scott, R. (2007). The mathematical theory of finite element methods (Vol. 15). Springer Science & Business Media.
10. Oden, J. T., & Reddy, J. N. (2012). An introduction to the mathematical theory of finite elements. Courier Corporation.
11. R. Hirota, J. Phys. Soc. Jpn. 43 (1977) 4116-4124.
12. R. Hirota, J. Phys. Soc. Jpn. 43 (1977) 2074-2078.
13. R. Hirota, J. Phys. Soc. Jpn. 43 (1977) 2079-2086.
14. R. Hirota, J. Phys. Soc. Jpn. 45 (1978) 321-332.
15. R. Hirota, J. Phys. Soc. Jpn. 46 (1979) 312-319.
16. M. J. Ablowitz and J. F. Ladik, J. Math. Phys. 16 (1975) 598-603.
17. M. J. Ablowitz and J. F. Ladik, J. Math. Phys. 17 (1976) 1011-1018.
18. M. J. Ablowitz and J. F. Ladik, Stud. Appl. Math. 55 (1977) 213-229.
19. M. J. Ablowitz and J. F. Ladik, Stud. Appl. Math. 57 (1977) 1-12.
20. T. R. Taha and M. J. Ablowitz, J. Comput. Phys., 55 (1984), 192-202.
21. T. R. Taha and M. J. Ablowitz, J. Comput. Phys., 55 (1984), 203-230.
22. T. R. Taha and M. J. Ablowitz, J. Comput. Phys., 55 (1984), 231-253.
23. T. R. Taha and M. J. Ablowitz, J. Comput. Phys., 55 (1988), 540-548.
24. Hairer, E., Lubich, C., & Wanner, G. (2006). Geometric numerical integration: structure-preserving algorithms for ordinary differential equations. Springer Science & Business Media.
25. Gonzalez, O., Time integration and discrete Hamiltonian systems, J. Nonlinear Sci., 6 (1996), 449‐467.
26. Greenspan, D., Discrete Models, in Applied Mathematics and Computation, Addison‐Wesley Publishing Co., Reading, 1973.
27. 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.
28. 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).
29. Furihata, D., & Matsuo, T. (2010). Discrete variational derivative method: a structure-preserving numerical method for partial differential equations. Chapman and Hall/CRC.
30. 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.
31. Bramble, 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.
32. Matsunaga, N., & Yamamoto, T. (2000). Superconvergence of the Shortley–Weller approximation for Dirichlet problems. Journal of Computational and Applied Mathematics, 116(2), 263-273.
33. 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
34. Swarztrauber, 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.
35. 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.
36. 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.
37. Fang, Q. (2006). Convergence of Ascher-Mattheij-Russell Finite Difference Method for a Class of Two-point Boundary Value Problems. INFORMATION, 9(4), 563.
38. 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.
39. M. Nakao, M. Plum, Y. Watanabe (2019) Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations (Springer Series in Computational Mathematics).
40. 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.
41. Gidas, B., Ni, W. M., & Nirenberg, L. (1979). "Symmetry and related properties via the maximum principle." Communications in Mathematical Physics, 68(3), 209–243.
42. Takayasu, A., Yoon, S., & Endo, Y. (2019). Rigorous numerical computations for 1D advection equations with variable coefficients. Japan Journal of Industrial and Applied Mathematics, 36(2), 357-384.
43. Cyranka, J. (2015). Existence of globally attracting fixed points of viscous Burgers equation with constant forcing. A computer assisted proof. Topological Methods in Nonlinear Analysis, 45(2), 655-697.
44. Cyranka, J., & Zgliczynski, P. (2015). Existence of globally attracting solutions for one-dimensional viscous Burgers equation with nonautonomous forcing-A computer assisted proof. SIAM Journal on Applied Dynamical Systems, 14(2), 787-821.
45. A. Takayasu, M. Mizuguchi, T. Kubo, and S. Oishi: "Accurate method of verified computing for solutions of semilinear heat equations", Reliable computing, Vol. 25, pp. 74-99, July 2017.
46. Mizuguchi, M., Takayasu, A., Kubo, T., & Oishi, S. I. (2014, September). A sharper error estimate of verified computations for nonlinear heat equations. In SCAN 2014 Book of Abstracts (p. 119-120).
47. Mizuguchi, M., Kubo, T., Takayasu, A., & Oishi, S. (2013) A priori error estimate of inhomogeneous heat equations using rational approximation of semigroups. Japan Society for Simulation Technology.
48. Wilczak, D. (2003). Chaos in the Kuramoto–Sivashinsky equations—a computer-assisted proof. Journal of Differential Equations, 194(2), 433-459.
49. 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.
50. Zgliczynski, P. (2004). Rigorous numerics for dissipative partial differential equations II. Periodic orbit for the Kuramoto–Sivashinsky PDE—a computer-assisted proof. Foundations of Computational Mathematics, 4(2), 157-185.
51. Zgliczynski, P., & Mischaikow, K. (2001). Rigorous numerics for partial differential equations: The Kuramoto—Sivashinsky equation. Foundations of Computational Mathematics, 1(3), 255-288.
52. 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.
53. 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.
54. 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.
55. 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.
56. 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.
57. Numerical Methods for Partial Differential Equations, Wiley Online Library
58. Driscoll, T. A., Hale, N., & Trefethen, L. N. (2014). Chebfun guide.
59. 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.
60. Hashemi, B., & Trefethen, L. N. (2017). Chebfun in three dimensions. SIAM Journal on Scientific Computing, 39(5), C341-C363.
61. 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.
62. Hecht, F. (2012). New development in FreeFem++. Journal of numerical mathematics, 20(3-4), 251-266.
63. Hecht, F., Pironneau, O., Le Hyaric, A., & Ohtsuka, K. (2005). FreeFem++ manual.
64. Sadaka, G. (2012). FreeFem++, a tool to solve PDEs numerically. arXiv preprint arXiv:1205.1293.
65. 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).
66. 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.
67. 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.
68. Langtangen, H. P., Logg, A., & Tveito, A. (2016). Solving PDEs in Python: The FEniCS Tutorial I. Springer International Publishing.

• 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.