﻿ REU projects

# REU Projects

### at Northern Arizona University with John Neubrger and Jim Swift May 29 - July 20, 2012

Prerequisites: I want my students to have taken differential equations and linear algebra. It is preferable if the student has taken the upper division physics course in mechanics. All of these problems will require some programming, using either Mathematica, MATLAB, Java or C++.

### Projects in the Galerkin Newton Gradient Algorithm (GNGA)

Neuberger, Sieben, and Swift have developed an algorithm (the GNGA) for solving the Partial Differential Equations (PDE). The simplest example is the PDE (1)
Δu = s u + u3
for real valued or u(x, y) or u(x, y, z), where Δ is the Laplacian. The PDE depends on the parameter s, and we study how the solution branches bifurcate as the parameter is varied. The symmetry of the region in the x-y plane, or in x-y-z space, has a profound effect on the bifurcations.

We also study the Ordinary Differential Equation (ODE) for u(x), where the Laplacian is just the second derivative:

u'' = s u + u3

1. Use the GNGA to find stationary (time independent) solutions to the Swift-Hohenberg equation. This involves solving the PDE (2)

(1 + Δ)2 u = s u - u3
(The author Jack Swift has no relation to me.) This has a built-in length scale, so there are essentially 2 parameters, s and L (the size of the region). The boundary conditions are also important. The two simplest examples of this problem have a one-dimensional domain, -L/2 < x < L/2, with either periodic boundary condtions u(x + L) = u(x) for all x, or the boundary conditions u(± L/2) = u'(± L/2) = 0. In the periodic boundary boundary conditions case, start by restricting the function space to even functions, or to odd functions.

2. Use the polynomial speed-up for the PDE (1) or (2). This is is done on the square region in the original paper by Neuberger and Swift. Since that original paper, we have focused on algorithms that work for general nonlinearities. Our GNGA algorithm runs slowly on the cube and it could run much faster using this technique. Solve the PDE on the square or the interval with polynomial-speedup, and

3. Correctly handle the symmetry in solving the PDE (1) or (2) on the disk or circle. Modify our existing C++ code, or write a MATLAB program. The continuous symmetry of the circle gives some interesting challenges.

The folloing ideas are probably not going to be persued this summer.

3. Do the GNGA with using a Quasi-Newton's method in place of Newton's method. This has the possibility of greatly speeding up the algorithm, especially for PDEs on 3-dimensional regions.

5. Solve the PDE (1) on the surface defined by x4 + y4 + z4 = 1. This surface has the symmetry of the cube, but the region is 2-dimensional.