Discretization can be understood as replacement of an exact solution of a PDE or a system of PDEs in a continuum domain by an approximate numerical solution in a discrete domain. Instead of continuous distributions of solution variables we find a finite set of numerical values that represent an approximation of the solution.
The spectral methods are similar to the technique of separation of variables used to solve PDE analytically.
The principal difference is that the series is infinite in the analytical method, and we try to find the coefficients such that the
series converges to the exact solution of the PDE. In the numerical spectral method, the series is finite and the coefficients are chosen so that they minimize the error of approximation
Finite Element Methods
The finite element methods are widely applied in many areas of engineering—for example, in structural analysis and conduction heat
transfer. They are also used for simulating fluid flows, although not as widely as the finite difference and finite volume methods. The basic concept is similar to that of the spectral methods. The main difference is that the decomposition is done not in the entire solution domain but within each of many small elements, into which the domain is subdivided. A small number of trial functions is used. The functions are chosen so that they are zero outside the element.
Finite Difference and Finite Volume Methods
The discretization is achieved by approximating the continuous solution at discrete grid points and time layers of a computational grid.
The domain of solution is covered by a grid of points with coordinates (x, y, z )i, where the index i is used to number the points.
If a marching problem is solved, the time range [t(0), t(end)] should also be covered by discrete points (time layers) t^n
Goal is to find the set of numbers u(i) that approximates the exact solution u(x, t ) at points (x, y, z )i and layers t^n . This is achieved by
approximating the original PDE using finite differences and solving the resulting system of algebraic equations.