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A method is presented for constructing finite difference schemes of high accuracy for ordinary differential equations. Since it is based on power series representation of the solution it is applicable to partial differential equations as well. The method is demonstrated on the general nonlinear first order ordinary differential equation and the truncation errors are analysed. A-stability of the method is analysed and it is found to be stable in this sense to arbitrarily high order. Finally a nonlinear example is presented. (Author)

Self-focussing and self-trapping of light rays are studied in a classical two-dimensional nonlinear model both in the geometrical optics approximation and when retaining higher dispersion terms. The paraxial ray approximation of Akhmanov that leads to the nonlinear Schrodinger equation is specifically avoided. It is found that in the geometrical optics approximation, focussing must eventually occur once convergence of rays begins. When both diffraction and nonlinearity are present, however, focussing may or may not occur. (Author)

The power series method is applied to the heat equation. Highly accurate semi-discrete systems of equations in t and in x are generated and are made stable by proper choice of parameters. A totally discrete scheme is produced that represents arbitrarily high accuracy in both x and t. Stability analysis indicates that while arbitrary order in t may be stable, the order of accuracy in x is restricted to be less than 16 and certain geometrical restrictions on the step sizes must be met. Truncation errors are examined and a consistency condition is obtained that further restricts the step sizes. The scheme is shown to coincide with Keller's Box scheme in its lowest order.

The power series method used to generate highly accurate finite difference schemes for ordinary differential equations is here applied to the wave equation. The analysis involves semi-discrete approximations in t and in x before the totally discrete scheme is derived. The results differ in that an arbitrarily accurate difference scheme is found for the wave equation that is stable and consistent with the differential equation. No such scheme exists for the heat equation. The step sizes in x and t must be equal for this difference scheme. Other difference schemes that do not restrict the step sizes are stable only when the order of accuracy in x is less than 5. The lowest order scheme is shown to coincide with Keller's Box Scheme.