Finite-difference method for evolution PDEs

Read the article , we discussed the numerical solution of stationary partial differential equation problems using the finite difference method. This method can be extended to the solution of evolution problems. We will study two types of problem: firstly, first-order evolution problems in time, also known as parabolic problems and, secondly, second-order evolution problems in time, also known as hyperbolic problems . The equations involved in these problems consist partly of a combination of partial derivatives with respect to the temporal variable, the numerical treatment of which we shall describe in detail, and partly of a combination of partial derivatives with respect to the spatial variable; the latter part was dealt with in detail in the article , the problem can be posed in a domain Ω, one-dimensional, two-dimensional or three-dimensional; to simplify the presentation we'll consider the domain to be the segment [0, 1], the two- and three-dimensional case presenting no major difficulties.