Numerical partial differential equations is the branch of numerical analysis

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....

 that studies the numerical solution of partial differential equations (PDEs).

Numerical techniques for solving PDEs include the following:

• The finite difference method
  Finite difference method
  In mathematics, finite-difference methods are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives.- Derivation from Taylor's polynomial :...
  
  , in which functions are represented by their values at certain grid points and derivatives are approximated through differences in these values.
• The method of lines
  Method of lines
  The method of lines is a technique for solving partial differential equations in which all but one dimension is discretized. MOL allows standard, general-purpose methods and software, developed for the numerical integration of ODEs and DAEs, to be used...
  
  , where all but one variable is discretized. The result is a system of ODEs in the remaining continuous variable.
• The finite element method
  Finite element method
  The finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...
  
  , where functions are represented in terms of basis functions and the PDE is solved in its integral (weak) form.
• The finite volume method
  Finite volume method
  The finite volume method is a method for representing and evaluating partial differential equations in the form of algebraic equations [LeVeque, 2002; Toro, 1999]....
  
  , which divides space into regions or volumes and computes the change within each volume by considering the flux (flow rate) across the surfaces of the volume.
• The spectral method
  Spectral method
  Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain Dynamical Systems, often involving the use of the Fast Fourier Transform. Where applicable, spectral methods have excellent error properties, with the so called "exponential...
  
  , which represents functions as a sum of particular basis functions, for example using a Fourier series
  Fourier series
  In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
  
  .
• Meshfree methods
  Meshfree methods
  Meshfree methods are a particular class of numerical simulation algorithms for the simulation of physical phenomena. Traditional simulation algorithms relied on a grid or a mesh, meshfree methods in contrast use the geometry of the simulated object directly for calculations. Meshfree methods exist...
  
   don't need a grid to work and so may be better suited for some problems. However the computational effort is usually higher.
• Domain decomposition methods
  Domain decomposition methods
  ]In mathematics, numerical analysis, and numerical partial differential equations, domain decomposition methods solve a boundary value problem by splitting it into smaller boundary value problems on subdomains and iterating to coordinate the solution between adjacent subdomains...
  
   solve boundary value problems by splitting them into smaller boundary value problems on subdomains and iterating to coordinate the solution between the subdomains.
• Multigrid method
  Multigrid method
  Multigrid methods in numerical analysis are a group of algorithms for solving differential equations using a hierarchy of discretizations. They are an example of a class of techniques called multiresolution methods, very useful in problems exhibiting multiple scales of behavior...
  
  s solve differential equations using a hierarchy of discretizations.

The finite difference method is often regarded as the simplest method to learn and use.
The finite element and finite volume methods are widely used in engineering

Engineering

Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

 and in computational fluid dynamics

Computational fluid dynamics

Computational fluid dynamics, usually abbreviated as CFD, is a branch of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows. Computers are used to perform the calculations required to simulate the interaction of liquids and gases with...

, and are well suited to problems in complicated geometries.
Spectral methods are generally the most accurate, provided that the solutions are sufficiently smooth.

See also

• List of numerical analysis topics#Numerical partial differential equations
• Numerical ordinary differential equations
  Numerical ordinary differential equations
  Numerical ordinary differential equations is the part of numerical analysis which studies the numerical solution of ordinary differential equations...
  
  

External links

• Numerical Methods for Partial Differential Equations course at MIT OpenCourseWare
  MIT OpenCourseWare
  MIT OpenCourseWare is an initiative of the Massachusetts Institute of Technology to put all of the educational materials from its undergraduate- and graduate-level courses online, partly free and openly available to anyone, anywhere. MIT OpenCourseWare is a large-scale, web-based publication of...
  
  .
• IMS, the Open Source IMTEK Mathematica Supplement (IMS)