Math 116: Numerical methods for PDEs and waves

2D scattering from sound-soft obstacle

Homework Resources Syllabus Student websites Final projects

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

• Week 1: Overview linear PDEs. Linear systems, norms, singular value decomposition. [NLA] Ch. 1-5.
  Worksheets: singvals.
  Codes:
  • vandermonde: shows numerical rank of said matrix which we proved was full rank.
  • ellipse: shows action of 2x2 matrix on unit ball
  • Jon's X-hr: show_taylor_error which uses the function taylor_error
• 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 q_n+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.
  
  • Jon's X-hour codes: intsp, fib2, matrixfill, sp.
• 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:
  • Code for HW4 #3 c, which people struggled with: hw4ans3c_nystrom_interpolation. Please read this through (it's short).
• 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.
  • Jon's X-hour codes: threeplt2
• 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.

Student homework websites

• Ben Chapman: 1 (codes), 2, 3 (probs 7, 8), 4
• Dan Fontaine
• Paul D. Nation
• Zaven Ovanesyan: 1, 2, 3, 4 (written part), 5, 6.
• Yong Su

Final student projects

• Ben Chapman: Transmission Scattering
• Dan Fontaine: Adaptive Interior Laplace BVP
• Paul D. Nation: Simulating a Single-Port Driven Quantum Dot (archive containing codes and write-up)
• Zaven Ovanesyan: Induced Charges on a Dielectric Sphere in a Constant Electric Field
• Yong Su: Capacity of the Unit Square