High-order finite difference methods for partial differential equations

General n-point formulae for difference operators and their errors are derived in terms of elementary symmetric functions. These are used to derive high-order, compact and parallelisable finite difference schemes for the decay-advection-diffusion and linear damped Korteweg-de Vnes equations. Stability calculations are presented and the speed and accuracy of the schemes is compared to that of other finite difference methods in common use. Appendices contain useful tables of difference operators and errors and present a stability proof for quadratic inequalities. For completeness, the appendices conclude with the standard Thomas method for solving tri-diagonal systems.