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:

• Symplectic integrators^[24]
• Discrete gradient method^{[25][26][27][28]}
• Discrete variational derivative method (DVDM)^[29]

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:

• Shortley-Weller approximation^[31][32][33]
• Swarztrauber-Sweet approximation^[34][35][36]
• Ascher-Mattheij-Russell difference formula^[37][38]

Validated Numerics for PDEs

See also: Validated numerics

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]

PDEs that have been Studied in the Context of Validated Numerics

• Advection equation^[42]
• Burgers equation^[43][44]
• Heat equation^[45][46][47]
• Kuramoto–Sivashinsky equation^{[48][49][50][51]}
• Navier-Stokes equations^[52][53]
• Orr–Sommerfeld equation^[54][55][56]

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:

• FreeFem++^[62][63][64]
• FEniCS Project^{[65][66][67][68]}

Experts in this Field

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