Numerical convective schemes based on accurate computation of space derivatives

A numerical procedure is developed for the solution of partial differential equations for the convection of scalar variables. An important feature of this approach is that the space derivatives are computed with very high accuracy by means of Fourier transform methods. By this method a forward marching problem involves discrete time steps but space derivatives are accurate within the limit to which a distribution can be defined on a finite set of meshpoints. Expressions are derived for the amplitude error and phase error, which are verified by computer experiments for the two-dimensional case. This numerical method is applied to the solution of Burgers' equation. The accuracy of the numerical solutions indicates that the method is well suited for the solution of nonlinear partial differential equations with other than periodic boundary conditions.