We will go through the notebook $UWHPSC/homeworks/project/BVP.ipynb, also visible at http://nbviewer.ipython.org/url/faculty.washington.edu/rjl/notebooks/BVP.ipynb. This outlines a recursive domain decomposition approach to solving a boundary value problem. Part 1 of the project is to convert this into Fortran with OpenMP.

Working in pairs, copy this notebook to BVP2.ipynb and modify it to solve a Helmholtz equation (in one dimension) of the form:

\(u''(x) + k^2 u(x) = -f(x)\)

on the interval \(0<x<1\) with specified boundary conditions.

As an exact solution, consider the case \(f(x)=0\) in which case the general solution to \(u''(x) = -k^2 u(x)\) is \(u(x) = c_1 \cos(kx) + c_2 \sin(kx)\).

The boundary value problem has a unique exact solution for any boundary values \(u(0)\) and \(u(1)\) provided that \(k\) is not an integer multiple of \(\pi\). (Insert \(x=0\) and \(x=1\) into the general solution and determine \(c_1\) and \(c_2\) so that the boundary conditions are satisfied.)

You might try values such as:

k = 15. u_left = 2. u_right = -1.

You will need to use at least 40 or 50 grid points to get a solution that looks at all reasonable. If you make \(k\) larger, the solution will be more oscillatory and you will need even more grid points to get a reasonable approximation.

Work through as much of the notebook as you can, adjusting things to solve the Helmholtz equation. The main objective is to work through the notebook and understand what is being done.

Some tips:

Add another parameter k to the solve_BVP_* functions.

In setting up the tridiagonal matrix in solve_BVP_direct, you will need to modify the diagonal terms for the difference equation that approximates the Helmholtz equation,

\(\frac{U_{i-1} - 2U_i + U_{i+1}}{\Delta x^2} + k^2 U_i = -f(x_i)\)

This gives the linear system to be solved:

\(U_{i-1} + (k^2\Delta x^2 - 2) U_i + U_{i+1} = -\Delta x^2 f(x_i)\)

along with the boundary conditions.

If you get to the divide-and-conquer approach, you will have to modify the equation for the mismatch to take into account the modification to the linear system being solved.

There is now a sample solution in the repository, visible at http://nbviewer.ipython.org/url/faculty.washington.edu/rjl/notebooks/BVP_helmholtz.ipynb.

**There is quiz for Lab 18**