There are four common numerical methods used to solve problems in mathematical physics.

Finite Difference Method

Solves differential equations by approximating derivatives with finite differences. The Taylor Series expansion is important to the method as it converts the differential equations to algebraic equations. This method is often taught in university courses on numerical methods since it can be easily programmed to solve problems.

Finite Volume Method

It uses Gauss's Divergence Theorem to convert volume integrals to surface integrals. It basically states that any change in the volume must be due to the net flow through the bounding surfaces of the volume. Partial Differential Equations (PDE's) are converted to algebraic equations and then solved. Most computational fluid dynamics (CFD) codes use this numerical method.

Boundary Element Method

A method for solving linear PDE's which have been written as integral equations. In this method, we are only concerned with solving the problem on the outer boundary. This method is often used for acoustics, electromagnetic problems and fracture mechanics. For problems with small surface/volume ratio, this method is highly efficient in terms of computational resources.

Finite Element Method

It is a very versatile method that can be used to solve PDE's in two or three space variables (ie: some boundary value problems) A continuous domain is discretized (meshed) into finite elements. The simple algebraic expressions used for each element are then combined to form a set of algebraic equations to be solved that represents the mesh. The solutions obtained are approximate and it is up to the user to validate and verify the results obtained.