Numerical techniques for solving PDEs include the following:
The finite difference method, 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, where functions are represented in terms of basis functions and the PDE is solved in its integral (weak) form.
The finite volume method, 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, which represents functions as a sum of particular basis functions, for example using a Fourier series.
Meshfree methods don't need a grid to work and so may be better suited for some problems. However the computational effort is usually higher.
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.
