1. MA5251: Spectral Methods & Applications
Weizhu Bao
Department of Mathematics
& Center for Computational Science and Engineering
National University of Singapore
URL:

2. Contents Introduction and Preliminaries Spectral-Collocation Methods
Some numerical examples
Review of different numerical methods of PDE
Historical background of spectral methods
Some examples of spectral methods
Fourier series and orthogonal polynomials
Review of iterative solvers and preconditioning
Review of time discretization methods
Spectral-Collocation Methods
Introduction
Differentiation matrices
Fourier, Chebyshev collocation methods

3. Contents Spectral-Galerkin methods
Introduction
Fourier spectral method
Legendre spectral method
Chebyshev spectral method
Error estimates
Spectral methods in unbounded domains
Hermite spectral method
Laguerre specreal methods,….

4. Contents Applications In fluid dynamics In heat transfer
In material sciences
In quantum physics and nonlinear optics
In plasma and particle physics
In biology
…………..

5. Dynamics of soliton in quantum physics

6. Wave interaction in plasma physics

7. Wave interaction in plasma physics

8. Wave interaction in plasma physics

9. Wave interaction of plasma physics

10. Wave interaction in particle physics

11. Vortex-pair dynamics in superfluidity

12. Vortex-dipole dynamics in superfluidity

13. Vortex lattice dynamics in superfluidity

14. Vortex lattice dynamics in superfluidity

15. Vortex lattice dynamics in BEC

16. Main numerical methods for PDEs
Finite difference method (FDM) – MA5233
Advantages:
Simple and easy to design the scheme
Flexible to deal with the nonlinear problem
Widely used for elliptic, parabolic and hyperbolic equations
Most popular method for simple geometry, ….
Disadvantages:
Not easy to deal with complex geometry
Not easy for complicated boundary conditions
……..

17. Main numerical methods
Finite element method (FEM) – MA5240
Advantages:
Flexible to deal with problems with complex geometry and complicated boundary conditions
Keep physical laws in the discretized level
Rigorous mathematical theory for error analysis
Widely used in mechanical structure analysis, computational fluid dynamics (CFD), heat transfer, electromagnetics, …
Disadvantages:
Need more mathematical knowledge to formulate a good and equivalent variational form

18. Main numerical methods
Spectral method – This module
High (spectral) order of accuracy
Usually restricted for problems with regular geometry
Widely used for linear elliptic and parabolic equations on regular geometry
Widely used in quantum physics, quantum chemistry, material sciences, …
Not easy to deal with nonlinear problem
Not easy to deal with hyperbolic problem
…..

19. Main numerical methods
Finite volume method (FVM) – MA5250
Flexible to deal with problems with complex geometry and complicated boundary conditions
Keep physical laws in the discretized level
Widely used in CFD
Boundary element method (BEM)
Reduce a problem in one less dimension
Restricted to linear elliptic and parabolic equations
Need more mathematical knowledge to find a good and equivalent integral form
Very efficient fast Poisson solver when combined with the fast multipole method (FMM), …..

20. Historical background
Method of weighted residuals (MWR) – Finlayson & Scriven (1966)
Trial functions (or expansion or approximation functions): are used as the basis functions for a truncated series expansion of the solution.
Test functions (or weight functions): are used to ensure that the differential equation is satisfied as closely as possible by the truncated series expansion.
This is achieved by minimizing the residual, i.e. the error in the differential equation produced by using the truncated expansion instead of the exact solution, with respect to a suitable norm.
An equivalent requirement is that the residual satisfy a suitable orthogonality condition with respect to each of the test functions.

21. Historical background
Trial functions
Spectral method: infinitely differentiable global functions, i.e. eigenfunctions of singular Sturm-Liouville problems
Finite Element Method (FEM): partition the domain into small elements, and a trial function (usually polynomial) is specified in each element and thus local in character & well suited for handling complex geometries.
Finite Difference Method (FDM): similar as FEM.
Test functions
Spectral methods: three different ways
FEM: similar as trial functions
FDM: Dirac delta functions centered at the grid points

22. Historical background
Different test functions of spectral methods
Galerkin method: same as the trial functions which are infinitely smooth functions & individually satisfy the boundary conditions. The differential equation is enforced by requiring that the integral of the residual times each test function be zero.
Collocation method: Dirac delta functions centered at the collocation points. The differential is required to be satisfied exactly at the collocation points.
Spectral tau method: Similar as the Galerkin method except that no need the trial and test functions satisfy the boundary conditions. A supplementary set of equations is used to apply the boundary conditions.

23. Historical background
Collocation approach (simplest of the MWR) – Slater (1934); Kantorovic (1934); Frazer, Jones and Skan (1937).
Proper choice of trial functions and distribution of collocation points – Lanczos (1938)
Orthogonal collocation method – Clenshaw (1957); Clenshaw and Norton (1963); Wright (1964).
Earliest application of spectral methods to PDE – Kreiss and Oliger (1972)—Fourier method; Orszag (1972) –pseudospectral.
Spectral-Galerkin method – Silberman (1954) in meteorological modeling; Orszag (1969, 1970); etc.

24. Historical background
Theory of spectral method -- Gottlieb and Orszag (1977)
Symposium Proceedings – Voigt, Gottlieb and Hussaini (1984)
First International Conference on Spectral and High Order Methods (ICOSAHOM) -- Como, Italy in It becomes series conference every three years. The next one is