A Course in Scientific Simulation Mike O’Leary Shiva Azadegan Towson University Supported by the National Science Foundation under grant DUE

What is the Course? This is a one-semester interdisciplinary course straddling the boundaries between mathematical modeling, numerical methods, and modern object-oriented computer programming. Our course is project-driven. Given a realistic problem, we  Create a model,  discuss appropriate numerical methods, and then  create a simulation of the problem using Microsoft Visual C++ that takes full advantage of our computer's graphical capabilities. Prerequisites: Calculus 1,2; Introduction to Programming 1.

How does the Course Proceed? The course is driven by the projects. The modeling, numerical methods, and computer programming are introduced as needed for the solution of a particular problem. Course can be split into four large portions  Introduction  First project  Second project  Third project

What are the Projects? Each project takes students 2-4 weeks to complete. Students complete a written project report  pages  Describes in detail the model, the numerical methods, and the program used to solve the problem.  Gives a complete answer to the assigned project questions. Students programs are also graded.

What have we developed? Lecture notes for the entire course CD that contains source code for the examples. These are available here, and online at

Motion of a baseball under air resistance What is the optimal angle to hit a baseball so that it travels the farthest? Does this angle change with the velocity of the ball? Students write a program that gives the total distance the ball will travel. This project does not require a graphical component, but one can be incorporated.

The double spring Describe the motion of a mass attached to two springs in the plane. Project questions include:  Given initial conditions, determine the location of the mass after a period of time, and estimate the accuracy of the answer.  Are there solutions that remain above the line connecting the two fixed points for all time?  Analyze the stability properties of the system. Does it display sensitive dependence on initial conditions?  Prove that the sum of the kinetic energy and the energy stored in the springs is conserved.

The resonant filter Create a model of an LRC resonant filter. Project questions:  Given an unknown input signal known to be the sum of sinusoids, find the frequency of the unknown signal. This requires an analysis of the preferred frequency of the solutions to a constant coefficient second order differential equation. Programming notes  The interface is best built with slider controls.  Because this requires displaying a graph of a function, some more complex graphical programming is needed.  Almost requires the use of dynamically created arrays.

Dynamics of HIV Create a simple one-compartment model for HIV infection of CD4+ T-cells. Obtain three equations in three variables. Investigate the effect of an RT-inhibitor. Though they do not kill HIV, they prevent HIV cells from infecting healthy cells. Project questions  Show that that the system tends to a steady state.  Show that, if the RT inhibitor is sufficiently effective, then HIV will be entirely removed from the body.  Show that, there is a lower threshold for effectiveness of the RT inhibitor which, though it would not eliminate the virus, it would prevent the onset of AIDS.  Compare the numerical results with their analytic counterparts.

The double pendulum Create a model for a jointed pendulum. Best technique requires variational methods. Project questions  Does the double pendulum display sensitive dependence on initial conditions?

Diffusion Create a model for diffusion processes. Three different methods to create the model-  System of ordinary differential equations  Directly, as a partial differential equation  Probabilistic methods. Introduce finite difference methods for solving partial differential equations  Discuss consistency, stability and convergence. Project question:  Simulate the temperature of a solid bar. Programming note: the best way to enter the initial data is to use the mouse.