Numerical methods for partial differential equations
Numerical partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs).
Numerical techniques for solving PDEs
Finite difference method
Finite difference method
In this method, functions are represented by their values at certain grid points and derivatives are approximated through differences in these values.
Method of lines
In this method, all but one variable is discretized. The result is a system of ODEs in the remaining continuous variable.
Finite element method
In this method, functions are represented in terms of basis functions and the PDE is solved in its integral (weak) form.
Finite volume method
This method 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.
Spectral method
This method represents functions as a sum of particular basis functions, for example using a Fourier series.
Meshfree methods
These methods 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
These methods solve boundary value problems by splitting them into smaller boundary value problems on subdomains and iterating to coordinate the solution between the subdomains.
Multigrid methods
These methods 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 and in computational fluid dynamics, 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 methods for partial differential equations
- Numerical ordinary differential equations
External links
- Numerical Methods for Partial Differential Equations course at MIT OpenCourseWare.
- IMS, the Open Source IMTEK Mathematica Supplement (IMS)
- Numerical PDE Techniques for Scientists and Engineers, open access Lectures and Codes for Numerical PDEs
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