Suggested topics for the final presentation

  1. Modified equation. (Possible sources: Strikwerda, Chap. 5; Hoffmann, Secs. 10.3, 10.7, 10.9, 12.3-12.5; Griffiths & Higham, Chaps. 13-15). Illustrate it using the example of symplectc methods of Lec. 5.
  1. Explicit symplectic methods of order higher than 2; e.g., the Forest—Ruth algorithm and its generalization (
  1. Singular IVPs and BVPs (with nonsingular solutions, like the Bessel function). (Possible sources: Champine & Gordon’s book, Chap. 12; see also book by Bailey, Champine, Waltman; I have copies of both books.)
  1. Read the 1991 paper by N. Trefethen on pseudospectra of matrices:

and present a few examples from there. The idea is to demonstrate that there is “life beyond eigenvalues”.

  1. Two-component coupled parabolic PDEs: must use the block-Thomas algorithm.
  1. Equation governing vibrations of a bar under tension (probably with some dissipation - ??): u_tt + u_xxxx = u_xx. A method to solve it is described on p. 378 (#29) in the book by M. Jain, “Numerical solution of differential equations” (I have a copy).
  1. Model a rotating spiral in a nonlinear reaction-diffusion-type equation. (Possible source: paper by D. Barkley, “Spiral meandering in chemical waves and patterns”.)
  1. Model a stiff reaction-diffusion system with explanations of the choices of methods and parameters.
  1. Any of the 3 projects in the advanced notes by N. Kutz. The notes are found under the link “Useful links – online resources on numerical methods” on the course webpage. I also a have a copy of his notes.
  1. Solve a parabolic equation in 2D with a mixed-derivative term and time-dependent boundary conditions by the Graig—Sneyd method. Confirm accuracy and stability as described in Lecture 15. Possible sources: find a relevant Fokker—Planck (it occurs in the theory of random processes) equation in “Handbook of stochastic methods” by C. Gardiner (I have a copy).
  1. Read the paper by D. Duffy on Alternating-Direction-Explicit methods (I have a copy), especially its Eq. (30). The idea is to compare this approach with that presented in Lecture 15 and based on AD Implicit methods. You’ll need to verify accuracy and stability of Eq. (30) and apply that method to some 2D heat-like equation with time-dependent boundary conditions. D. Duffy is not a mathematician but an expert in mathematical finance, so the paper is written rather understandably.
  1. Read a paper by S. Chin on Saulyev-type methods (an alternative to ADI methods) and present his point of view on their stability. This is probably related to the previous topic (on ADE methods), but is independent of it.
  1. Read a report by R. LeVeque (I have a copy) on deriving the intermediate boundary conditions (as in Lec. 15, Sec. 6) using the concept of the modified equation (item 1 on this list).
  1. Read about discontinuous Galerkin methods. Explain why they are needed and their main idea (and possibly a simple example). Find the material online; it should be plentiful.
  1. Find information online about 1D diffusion (or diffusion-advection) equations with a moving boundary. Learn and present the basic idea and an example.
  1. Learn two finite-difference schemes for the wave equation u_tt = u_xx other than the simple central-difference scheme (e.g., Lax—Friedrichs, Lax—Wendroff, Beam—Warming, McCormack,…). Present their consistency and stability analyses.
  1. Relate item 1 (modified equation) to one of the schemes mentioned in the previous topic (f-d schemes for the wave equation).
  1. Chebyshev spectral methods. Possible sources: Trefethen’s book “Spectral methods in Matlab” and course notes by D. Chopp (see above). There can be two emphases: (i) How spectral methods can be used to time-efficiently compute derivatives of non-periodic functions, and (ii) How spectral methods can be used to solve PDEs.
  1. Stochastic differential equations. (Possible sources: Chopp’s notes; Riecke’s notes).Webpage of David Chopp:

Webpage of H. Riecke:

  1. Read Lecture 17 and apply its method to find the breather of Sine-Gordon (or Klein-Gordon) equation or the soliton of Aceves—Wabnitz equations (I can supply the equations).