A Unified FFT-Based Spectral Solver for High-Order PDEs (Orders 4-9) with Applications to Pattern Formation

Description

This research proposes the development of a unified and efficient solver for high-order partial differential equations (PDEs) of orders 4 through 9 by leveraging the inherent power of the Fast Fourier Transform (FFT). While traditional methods like finite differences become computationally prohibitive for such high orders due to the need for large stencils and extremely fine meshes, an FFT-based spectral approach offers a elegant solution by transforming derivative operations into simple algebraic multiplications in Fourier space, thereby treating a 9th-order derivative with the same ease as a 2nd-order one. The methodology will employ a pseudo-spectral framework, computing spatial derivatives via FFT and handling nonlinear terms in physical space, coupled with exponential time-differencing to manage the stiffness characteristic of high-order operators. This approach is expected to reduce computational complexity from linear scaling with the derivative order to (O(N log N)), making it feasible to simulate complex models on standard hardware. The solver's efficacy will be demonstrated through applications such as simulating the triharmonic equation (( Delta^3 u )) for elastic shell deformations and investigating generalized Swift-Hohenberg equations with 8th and 9th-order terms for pattern formation in fluid dynamics. By providing a "turn-key" tool that decouples physics from numerical complexity, this research will significantly advance the study of high-order phenomena, with its success ensured by the mathematical straightforwardness of spectral methods and the maturity of optimized FFT libraries.