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FD and Spectral methods for PDE
Finite differences and spectral methods for PDEs
1. Finite differences
1.1. Indefinite integration
1.2. Power series and finite differences
1.3. Roundoff error
1.4. Fourier analysis of finite differences
1.5. Interpolation and finite differences
1.6. Differentiation matrices
2. TPBVP
2.1. Two-point boundary-value problem
2.2. Boundary conditions for the TPBVP
2.3. Nonlinear TPBVP
2.4. Resolving layers
2.5. Bootstrapping methods
3. Elliptic PDE
3.1. Tensor-product grids
3.2. Solving the Poisson equation
3.3. Discretizing the disk
3.4. Iterative linear algebra
3.5. Domain decomposition
4. Method of lines
4.1. Multistep and RK methods
4.2. Absolute stability
4.3. Semidiscretization
4.4. Convergence theory
4.5. Beyond the basics
4.6. Multiple space dimensions
5. Advection
5.1. The CFL condition
5.2. von Neumann analysis
5.3. Group velocity and dispersion
5.4. Boundaries and interfaces
5.5. Hyperbolic systems
6. Spectral methods
6.1. Unbounded finite differences
6.2. Unbounded spectral
6.3. Periodic discrete functions
6.4. Convergence rates
6.5. Barycentric formula
6.6. Potential theory
6.7. Chebyshev differentiation matrix
6.8. Chebyshev for BVPs
6.9. Chebyshev and the FFT
6.10. Eigenvalues
6.11. Time stability
6.12. Polar coordinates
6.13. Spectral integration
6.14. Boundary conditions
6.16. Fourth-order PDEs
7. Advanced
7.1. Operator splitting
7.2. Radial basis functions
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This material is based on Trefethen’s book.