On a multiple scales perturbation method for partial difference equations

Abstract

In this paper, we introduce a multiple time scales (MTS) framework for partial difference equations (PΔEs). Such a framework is underdeveloped for fully discrete systems. We investigate a classical initial-boundary value problem for a PDE using a standard finite difference discretisation. For a nonlinear example, we additionally apply a nonstandard discretisation. Operators for fast and slow iteration scales are introduced, and secularity conditions governing the slow evolution of modal amplitudes are derived via discrete modal projection. Quantities such as natural frequencies and the stability of periodic solutions are analysed by comparing continuous and discrete MTS approximations. We prove the asymptotic validity of approximations in a Hilbert space setting. For standard discretisations, we derive the bounds on the mode numbers that can be accurately represented by given spatial and temporal resolutions. Beyond these bounds, the discrete natural frequencies can lead to spurious modal interactions, causing the approximations to fail at O(ε) accuracy over iteration scales of O1ε. For both standard and nonstandard schemes, we obtain qualitatively consistent solutions. Notably, the nonstandard discretisation yields solutions that exactly match the continuous PDE in the limit ε=0 and closely approximate the continuous MTS expansion for 0<ε≪1.