Spectral Methods and High-Order Numerical Methods 2026 xCSD
Spectral Methods and High-Order Numerical Methods Proposed 2026 English Course Outline Part I — Approximation and spectral representations Week 1 — Foundations of high-order discretization Model PDE classes: elliptic, parabolic, hyperbolic, dispersive Weighted-residual framework Galerkin, Petrov–Galerkin, tau, collocation and least-squares formulations Approximation error, consistency, stability and convergence Sobolev regularity versus spectral convergence Modal versus nodal representations Reproducible numerical experiments in Julia, Python or MATLAB Week 2 — Fourier approximation and the FFT Fourier series and transforms Truncation, interpolation and projection Parseval identities Trigonometric interpolation Discrete Fourier transform and FFT Spectral differentiation and integration Aliasing, convolution and the two-thirds/three-halves de-aliasing rules Gibbs phenomena and filtering Week 3 — Fourier methods for periodic PDEs Fourier–Galerkin and Fourier collocation methods Poi...