Numerical partial differential equations

Numerical partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs).

In this method, functions are represented by their values at certain grid points and derivatives are approximated through differences in these values.

The method of lines (MOL, NMOL, NUMOL) is a technique for solving partial differential equations (PDEs) in which all but one dimension is discretized. MOL allows standard, general-purpose methods and software, developed for the numerical integration of ordinary differential equations (ODEs) and differential algebraic equations (DAEs), to be used. A large number of integration routines have been developed over the years in many different programming languages, and some have been published as open source resources.

The method of lines most often refers to the construction or analysis of numerical methods for partial differential equations that proceeds by first discretizing the spatial derivatives only and leaving the time variable continuous. This leads to a system of ordinary differential equations to which a numerical method for initial value ordinary equations can be applied. The method of lines in this context dates back to at least the early 1960s.

The finite element method (FEM) is a numerical technique for finding approximate solutions to boundary value problems for differential equations. It uses variational methods (the calculus of variations) to minimize an error function and produce a stable solution. Analogous to the idea that connecting many tiny straight lines can approximate a larger circle, FEM encompasses all the methods for connecting many simple element equations over many small subdomains, named finite elements, to approximate a more complex equation over a larger domain.

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