Numerical schemes and difference equations

In this study we investigate the connection of difference equations and numerical schemes through the study of a simple partial differential equation (pde). After an introduction to different numerical schemes, we use some well known finite differences schemes to discretize the simple linear pde ut+2ux = 0 with initial condition (x,0) = x. The different discretization schemes lead to different, consistent to the original pde, numerical schemes which constitute corresponding partial difference equations. The solution of the above mentioned pde is attained numerically as well as by analytic solution of the corresponding difference equations. The results show that the solution is always attained by using the analytic solution of difference equations where as, limitations should be taken into consideration when we try to achieve the solution numerically. These results indicate that the analytic solution of difference equations, resulting from application of numerical schemes, could be of extreme importan ce for the estimation of the solution of a pde.

Keywords: analytic solution, difference equations, differential equations, basic theory of numerical schemes