These equations may depend on time and on one, two or three space dimensions. In such circumstances, explicit exact solutions will remain the dream of the mathematician for the foreseeable future. The differential equations governing the behaviour of an inviscid gas, the Euler equations, have been known to scientists for centuries, but the exact solutions of these equations available today are only valid for very simple physical situations. Exact solutions, even for unrealistic cases, are also helpful in assessing numerical methods intended for use in more general situations. Moreover, there are numerical methods that rely on exact solutions to the equations under simplified initial conditions assumed to hold locally.
Numerical methods are techniques for finding approximate solutions to mathematical problems.
The finite difference method, one of many methods available, is today capable of producing numerical solutions to both ordinary and partial differential equations. Real progress on the development and application of numerical methods is the product of the 20th century, the beginning of which, witnessed increased efforts by scientists to invent, analyse and apply numerical methods to solve differential equations that represented mathematical models of physical phenomena. Early contributions of historical importance were made by scientists such as Runge, Richardson, Liebmann, Courant, Friedrichs, von Neumann, amongst many others. Back to top Progress on numerical methods was made only when two basic ingredients were available, namely, numerical algorithms, along with a better understanding of their mathematical properties such as stability, and computing machines. The first ingredient has given rise to a new branch of Mathematics, namely Numerical Analysis.