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Numerical partial differential equations

 

 
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Numerical partial differential equations
 
 
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Numerical partial differential equations is the branch of numerical analysis
Numerical analysis

Numerical analysis is the study of algorithms for the problems of continuous mathematics .One of the earliest mathematical writings is the Babylonian tablet YBC 7289, which gives a sexagesimal numerical approximation of , the length of the diagonal in a unit square....
 that studies the numerical solution of partial differential equations (PDEs).

Numerical techniques for solving PDEs include the following:






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Numerical partial differential equations is the branch of numerical analysis
Numerical analysis

Numerical analysis is the study of algorithms for the problems of continuous mathematics .One of the earliest mathematical writings is the Babylonian tablet YBC 7289, which gives a sexagesimal numerical approximation of , the length of the diagonal in a unit square....
 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....
    , 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, 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 analysis for finding approximate solutions of partial differential equations as well as of 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 as 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 partial differential equations , often involving the use of the Fast Fourier Transform....
    , which represents functions as a sum of particular basis functions, for example using a Fourier series
    Fourier series

    In mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions, namely sine wave . The study of Fourier series is a branch of Fourier analysis....
    .
  • Meshfree methods
    Meshfree methods

    Meshfree methods are a particular class of numerical analysis 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....
     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 the subdomains....
     solve a 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. The idea is similar to extrapolation between coarser and finer grids....
    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 and profession of applying Technology and science knowledge and utilizing natural laws and physical resources in order to design and implement materials, structures, machines, devices, systems, and process that safely realize a desired objective and meet specified criteria....
 and in computational fluid dynamics
Computational fluid dynamics

Computational fluid dynamics is one of the branches of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows....
, 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




External links


  • 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 Post-secondary education- and Quaternary education courses online, Public domain and Open access to anyone, anywhere, by the end of the year 2007....
    .
  • , the Open Source IMTEK Mathematica Supplement (IMS)