Transcript Numerical Methods for Generalized Zakharov System

MA5251: Spectral Methods & Applications
Weizhu Bao
Department of Mathematics
& Center for Computational Science and Engineering
National University of Singapore
Email: [email protected]
URL: http://www.math.nus.edu.sg/~bao
Contents
Introduction and Preliminaries
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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
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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
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In fluid dynamics
In heat transfer
In material sciences
In quantum physics and nonlinear optics
In plasma and particle physics
In biology
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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:
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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
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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/