MTH472/572: Numerical Methods for PDEs
Spectral and Pseudospectral Methods
Fall 2013, Larts Room 209 Group I, 12:30 - 1:45 PM
Department of Mathematics, UMass Dartmouth
14 Week Tentative Schedule
The tentative schedule of this course is following the table of contents of the textbook: Spectral Methods in MATLAB by Nick Trefethen. If time permits ( a bit too ambitious maybe), I will also insert additional materials such as Barycentric Lagrange Interpolation, Mapped Chebyshev Grid + Rational Interpolant, and techniques in implementing boundary conditions such as Fictitious Point Method (Fornberg), Resampling Method (Hale & Driscoll), and Spectral Penalty Method (Hesthaven's paper).
Approximation and Derivatives
- Week 1: Differentiation Matrices (Sept 5):
- Syllabus and mycourses.
- A brief history of spectral and pseudospectral methods.
- Unbounded domain, Half unbounded domain, Periodic domain, Bounded domain.
- Lagrange interpolation polynomial. Derivation of 3pt stencil FD weights.
- Upload HW Chap1 in mycourses
- Week 2: DM continues + Unbounded Grids: SDFT (Sept 10 & Sept 12):
- Differentiation matrix (3pt stencil) for non-periodic and periodic domain.
- Upload Fornberg's (Generating FD weights) paper to mycourses.
- Discuss p1.m. Upload HW Chap2 in mycourses.
- Unbounded domain, uniform sampling points, aliasing effect.
- Fourier and inverse Fourier Transform. (also its semidiscrete version).
- Deriving Sinc interpolant (and differentiation matrices) for unbounded discrete domain.
- Discuss p3.m and Gibbs Phenomenon.
- Upload DMSUITE (Weideman & Reddy) paper and discuss the unbounded DM case.
- Ask students to download DMSUITE from matlab files exchange and learn a bit to use it.
- Week 3: SDFT continues + Periodic Grids: DFT and FFT (Sept 17 & Sept 19):
- Bring students to the first session of CSCVR workshop.
- Deriving the Sinc interpolant for periodic domain.
- Form differentiation matrices from the periodic interpolant.
- Discuss MATLAB FFT (fft, ifft) and warn students about the arrangement of wave numbers.
- Show convergence of 1st derivatives for the FFT case.
- Discuss p4.m and p5.m.
- Upload HW Chap3 in mycourses.
- Week 4: Smoothness and Spectral Accuracy (Sept 24 & Sept 26):
- Bring students to the second session of CSCVR workshop.
- big O, small o, smoothness of u and decay of \hat{u} (thm 1)
- L-infinity, L-2 (space and norms). Bounded variation.
- Riemann-Lebesque lemma.
- Convolution of box functions.
- Thm 2,3,4. Discuss p7.m
- Week 5: Polynomial Interpolation and Clustered Grids (Oct 1 & Oct 3):
- Derive Barycentric Lagrange Interpolant.
- Mention Salzer's 1972, Schwarz's 1997, and Wang's 2012 papers regarding Barycentric
weights for Chebyshev, Equally-spaced, and Legendre points.
- Runge Phenomenon, Ellipse of analyticity.
- Derive potentials due to equally-spaced points.
- Upload HW Chap 4,5,6 in mycourses.
- Give an extra HW problem by modifying p9.m. Get rid of p=polyfit(x,u,N) and pp = polyval(p,xx),
and use Barycentric Lagrange interpolant instead. For Chebyshev's case, try to increase N.
- Week 6: Chebyshev Differentiation Matrices (Oct 8 & Oct 10):
- Derive Chebyshev differentiation matrices from Barycentric Lagrange Formulation.
- Discussing Berrut and Trefethen's paper.
- Overview of DMSUITE by Weidemann and Reddy.
- Practice using cheb.m from Trefethen's book.
- Practice calling chebdiff from DMSUITE.
- Practice how to obtain DM from chebfun.
- Watch out the order of x (from 1 to -1) vs (-1 to 1)
Time Independent and Depedent PDE Problems
- Week 7: Boundary Value Problems (Oct 15 & Oct 17):
- MATLAB jam session in class. All students are bring their laptops with MATLAB.
- Solve 1D Poisson equation. Dirichlet and Neumann BCs.
- Be familiar with Tensor Product Grid.
- Solve 2D Poisson equation.
- Week 8: Time-Stepping and Stability Regions (Oct 22 & Oct 23):
- Stability regions of popular time stepping.
- Polynomial R(z) (explicit) and Rational function R(z) (implicit)
- Plot stability regions. Use surf and contourf.
- Solve 1D advection equation.
- Plot eigenvalues scaled with time-step k. (lie inside stability region).
- Week 9: Eigenvalues and Pseudospectra (Oct 29 & Oct 31):
- MATLAB jam session in class.
- 2D Heat equation and 2D wave equation.
- Students complaints memory issues when creating kron(D2,I) + kron(I,D2). Use speye to create I.
- Instead of creating time-stepping codes from scratch, show students how to use MATLAB ode solver.
- Students request that the final project should be about solving Navier-Stokes (OMG, these students are tough!)
- Week 10: Chebyshev Series, Root Finding, and the FFT (Nov 5 & Nov 7):
- 3 equivalent settings: Fourier, Laurent, Chebyshev.
- Recurrence relations of Chebyshev Polynomials.
- Root finding of interpolant based on monomials. Eigenvalues of companion matrix.
- Root finding of Chebyshev interpolant. Eigenvalues of colleague matrix.
- Week 11: Integrals and Quadrature Formulas (Nov 19 & Nov 21):
- Clenshaw-Curtis Quadrature.
- Code along with students in class
- Upload Fredholm integral equation problem in mycourses.
- Upload 2D wave equation project in polar coordinates inm mycourses.
- Week 12: Fourth-Order Problems (Nov 12 & Nov 14):
- Implementing boundary conditions in chapter 14.
- 1D and 2D cases of Biharmonic equations with zero Dirichlet BCs.
- Upload 2D Navier-Stokes in stream function formulation project.
- Let students work in class to finish projects.
- Week 13: More about Boundary Conditions (Nov 26):
- Introduce Fornberg's Fictitious point method for handling multiple BCs.
- Do 1D boundary value problem.
- MATLAB jam session in class.
- Not able cover Resampling Method (Hale & Driscoll) and Spectral Penalty (Hesthaven's paper). Ask students
to come to office hours if they want to know more about this.
- Week 14: Still more about Boundary Conditions (Dec 3 & Dec 5):
- Map Chebyshev points such that 2 last points become fictitious points.
- Create differentiation matrices in the mapped Chebyshev points.
- Solving 1D Biharmonic equation with non-zero multiple BCs and compare the numerical
solution with exact solution.
- Upload the MATLAB code to mycourses.
- Final Exam Week (Dec 11 - Dec 17): Projects Due.