An Introduction to the Spectral Method

Transport phenomena problems typically involve diffusion, convection and sometimes even radiation. Each of these are mathematically associated with space derivative operators. The derivative operator is of course expressed in “continuous” space. However, numerical representation of this continuous operator can only be achieved in “discretized” space, i.e., at a given number of grid points. We therefore begin by showing that the derivative operator, which is a linear operator, is represented by a matrix called the differentiation matrix. The grid definition as well as the way to obtain this matrix are given. We then tell the reader the way to take boundary conditions into account.