# Math 116: Numerical methods for PDEs and waves

## Alex Barnett, Fall 2008, Tu and Th 10:00-11:50am (10A), Kemeny 201

 2D scattering from sound-soft obstacle Rapid progress in computer power and numerical algorithms in recent decades has revolutionized science and technology. The Laplace equation (describing steady-state diffusion, heat flow, electrostatics) and Helmholtz equation (linear waves, acoustics, electromagnetics, optics, quantum) are linear PDE boundary value problems, ubiquitous in modeling the real world. They may be solved numerically by recasting the problem onto the boundary; this is more efficient at short wavelengths (and easier to code) than standard discretization methods. You will build codes, analyse their errors, and later explore phenomena in wave scattering and quantum chaos (short-wavelength asymptotics). You will learn some of the deep mathematics required to understand the success and efficiency of modern algorithms. Course Flyer

Admin: Office hours M 3-4pm, F 2-3 pm. Our course TA and coding coach is Jon Brown, who runs the X-hour 3pm Wed.
Consider the following Matlab classes, if you like to learn in a group (the 1st is a bit late to help, but the 2nd looks useful)

• Introduction to Matlab: Oct 8 4-6pm. Berry Instructional Center, Carson Hall.
• Programming With Matlab: Oct 16 4-6pm, Berry Instructional Center, Carson Hall.

### Lecture notes, topics, readings, worksheets, codes

Lecture notes are my own rough scribbles; brackets [NLA] etc refer to books, as listed in the syllabus document
• Week 1: Overview linear PDEs. Linear systems, norms, singular value decomposition. [NLA] Ch. 1-5.
Worksheets: singvals.
Codes:
• Week 2: Condition number, floating-point representation, stability. [NLA Ch. 12-14]. Interpolation on intervals. [NA] Ch. 8.1
Worksheets: interp.
Codes:
• charges_polynomial: shows contour lines of size of log of monic polynomial qn+1 in C for two node spacings
• charges_equilibrate: relaxation animation towards equilibrium for charges on an interval
• Week 3: Quadrature, convergence rate given by nearest singularity, error bounds and proving convergence [NA] Ch. 9.1-3.
Worksheets: gauss_n=2.
• Week 4: Periodic quadrature [NA] Ch. 8.2, 9.4. Integral Equations, Nystrom method. [Atkinson] Ch 1.1.-1.3, [NA] Ch. 12.3.
Codes:
• Week 5: Functional analysis and compact operators [NA] Ch. 12.1-2. Laplace's equation, Green's representation formula. [LIE] Ch. 6.1
Worksheets: laplace (Greens identities and checking the fundamental solution).
• Week 6: Potential theory, jump relations, boundary integral formulations. Interior and exterior Laplace BVP. [LIE] Ch. 6
• Code for HW6 #2,3: dlp_bvp_simple. Note the use of vectorization to fill Nystrom matrix with no loops.
• Week 7: Helmholtz scattering problems. Avoiding interior resonances: Brakhage-Werner-Leis-Panich method. [CK]. Dirichlet eigenvalues of domains, Method of Particular Solutions.
• Week 8: Error analysis of MPS. [Still, Barnett, Betcke papers] Minimax characterization of eigenvalues, Weyl's law (theorem) for eigenvalue counting asymptotics [GAR] Ch. 11.
Worksheets: eigbnds (baby Moler-Payne proof)
• Week 9: Low-rank approximation of kernels, Fast Multipole Method. In-class project presentations.
• Code for low-rank demonstration of 2D Laplace kernel: lowrank.

### Final student projects

Project topic ideas list. Final writeups due 9am Wed Dec 10. Course grades based on 60% HW, 40% project.