9.3 Spectral Methods

Spectral methods offer a powerful approach to solving differential equations, using global approximations for high accuracy. They excel with smooth solutions, leveraging Fourier series or polynomial expansions to represent functions across entire domains.

Implementation involves collocation or Galerkin techniques, transforming between physical and spectral spaces. Spectral methods provide exponential convergence for smooth problems but face challenges with discontinuities and complex geometries compared to finite difference or element methods.

Fundamentals of Spectral Methods

Principles of spectral methods

• Global approximation approach uses entire domain for approximation contrasts with local methods (finite differences)
• High-order accuracy achieves exponential convergence for smooth functions
• Spectral representation of functions expands solution in terms of basis functions
• Pseudospectral methods evaluate derivatives in physical space transform between spectral and physical spaces
• Spectral Galerkin methods solve weak form of equations in spectral space
• Suitable for problems with smooth solutions
• Efficient for certain geometries (rectangular domains, spherical shells)

Approximation with global basis functions

• Fourier series uses trigonometric basis functions $\sin(nx)$ and $\cos(nx)$ for periodic boundary conditions on uniform grid points
• Chebyshev polynomials use $T_n(x) = \cos(n \arccos(x))$ for non-periodic boundary conditions on non-uniform grid points (clustered at boundaries)
• Legendre polynomials serve as alternative polynomial basis orthogonal on $[-1, 1]$
• Expansion of solution: $u(x) \approx \sum_{n=0}^N a_n \phi_n(x)$ where $\phi_n(x)$ are basis functions and $a_n$ are coefficients
• Spectral differentiation performs exact differentiation of basis functions using matrix representation of differential operators

Implementation and Analysis

Implementation of spectral methods

• Collocation method enforces PDE at specific grid points transforms between physical and spectral spaces uses Fast Fourier Transform (FFT) for efficiency
• Galerkin method minimizes residual in weak form operates entirely in spectral space requires evaluation of inner products
• Choice of test functions:
  1. Collocation: Dirac delta functions
  2. Galerkin: Same as basis functions (orthogonality)
• Boundary conditions incorporate essential (Dirichlet) and natural (Neumann) conditions
• Time-dependent problems use method of lines approach and implicit-explicit (IMEX) schemes

Properties of spectral methods

• Spectral accuracy exhibits exponential convergence for smooth functions and algebraic convergence for non-smooth functions
• Stability analysis employs Von Neumann stability analysis for time-dependent problems and CFL condition for explicit time-stepping
• Aliasing errors caused by insufficient resolution of high frequencies addressed by dealiasing techniques (3/2 rule)
• Gibbs phenomenon produces oscillations near discontinuities mitigated by filtering techniques
• Error estimation uses residual-based error indicators and a posteriori error analysis

Spectral vs finite methods

• Accuracy: spectral methods provide highest accuracy for smooth solutions finite difference offers lower order accuracy finite element allows flexible accuracy with h- and p-refinement
• Flexibility in geometry: spectral methods limited to simple geometries finite difference uses structured grids finite element handles unstructured meshes and complex geometries
• Computational cost: spectral methods efficient for low to moderate dimensions finite difference has low cost per degree of freedom finite element incurs higher cost due to assembly
• Handling of discontinuities: spectral methods struggle (Gibbs phenomenon) finite difference employs shock-capturing schemes finite element uses adaptive refinement and discontinuous Galerkin
• Ease of implementation: spectral methods simple for basic problems finite difference straightforward finite element more complex due to mesh generation
• Parallelization: spectral methods' global nature limits scalability finite difference good for structured parallelism finite element suitable for domain decomposition