MA5251: Spectral Methods & Applications

Weizhu Bao

Department of Mathematics & Center for Computational Science and Engineering

National University of SingaporeEmail: bao@math.nus.edu.sg

URL: http://www.math.nus.edu.sg/~bao

Contents

Introduction and Preliminaries– 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

Contents

Spectral-Galerkin methods– Introduction– Fourier spectral method– Legendre spectral method– Chebyshev spectral method– Error estimates

Spectral methods in unbounded domains – Introduction– Hermite spectral method– Laguerre specreal methods,….

Contents

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

Dynamics of soliton in quantum physics

Wave interaction in plasma physics

Wave interaction in plasma physics

Wave interaction in plasma physics

Wave interaction of plasma physics

Wave interaction in particle physics

Vortex-pair dynamics in superfluidity

Vortex-dipole dynamics in superfluidity

Vortex lattice dynamics in superfluidity

Vortex lattice dynamics in superfluidity

Vortex lattice dynamics in BEC

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• ……..

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

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– …..

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), …..

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.

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

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.

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.

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 1989. It becomes series conference every three years. The next one is

http://www.math.ntnu.no/icosahom/