davidnkraemer/poisson1d

Implements the numerical solver to the 1D Poisson ODE

Poisson 1D numerical BVP solver

This is my final project for AMS 562.

TL;DR: Running the project

To run the driver program and generate the data, one can simply execute

$ make

To generate the data plots (given the existence of the data), one can simply execute

$ ./generate_plots

Description

The 1D Poisson equation is specified by -u'' = f on the interval [a,b], where f is continuous and u is the desired function to be solved. This project implements a finite differencing scheme for determining the function u. This scheme reduces to solving a special linear system Ax = b where A is symmetric, tridiagonal, and positive definite. The suite provided in the solvers library implements this scheme using three different matrix storage methods:

1. Square, dense matrices (LuSolver), which implement no data compression at all.
2. Packed upper triangular matrices (PpSolver), which exploits the symmetric property of A.
3. Packed symmetric tridiagonal matrices (TriDiagonal), which exploits both the symmetric and tridiagonal properties of A.

For any of these methods to work, the matrix A must be positive definite.

Driver program

With these distinct storage methods, the driver program solvers/main.cpp tests them on simple Poisson problems. These problems involve functions f which can be readily integrated in order to determine the exact solution u. For each sample function and for each storage method, it compares the exact solution with the computed solution according to the finite differencing scheme by computing the L2 and LInf norm errors between them. It also writes the computed and exact solutions to a set of files in the data directory.

Data visualization

The Python script generate_plots takes data generated from the driver program and plots the computed and exact solutions. For every problem with a specified function f and storage method, the resulting solutions are plotted in individual graphs. The L2 and LInf norm errors are displayed as well. All of these solutions are also plotted in an additional compiled figure.

Dependencies

The script requires a Python 3 interpreter and uses the following dependencies:

• numpy(==1.12.1)
• matplotlib(==2.0.2)

For prettier graphs, it pays to have the seaborn library installed as well.

Discussion

This scheme is startlingly accurate for these toy problems. I would like to find a more interesting function to generate data for, perhaps then to show the limits of finite differencing. But for these examples, the finite difference method is highly successful.