Numerical Methods

Now that we have derived a differential equation governing the behavior of the system, we would like to use various numerical methods to explore the possible practical uses of the model. We will explore two methods, Forward Euler’s method and the Runge-Kutta method.

Forward Euler’s Method
Forward Euler’s method is an iterative process that approximates a curve when a differential equation is known by evaluating the slope at the current point, "stepping" in that direction on the graph for a small step size, and then reevaluates the slope and repeats the process. Forward Euler’s method may be shown symbolically as

where is a function evaluated at and

Backward Euler’s Method
Backward Euler’s method is similar to Forward Euler’s method in the respect that it allows for simple "stepping" through the graph of a function based on a differential equation. However, the key difference is that Backward Euler’s method requires that the derivative function be evaluated at the new point instead of the old. Symbolically this is represented as

where is a function evaluated at and

Runge-Kutta Method
The Runge-Kutta method is a more advanced method of approximating curves given a differential equation. The equations governing the Runge-Kutta method are

When using the Runge-Kutta method with our discrete data, we must interpolate to find the and terms. We used a linear approximation of the and data as our method of interpolation.

Error Measurement and Method Accuracy
The convergence of the various methods of approximating a function when the derivative is known is measured by comparing the order of the methods. Both Forward Euler’s method and Backward Euler’s method have an order of 1, which means that the error is proportional to the step size . For example, if is made twice as small, the error in the resulting approximation will also be twice as small. However, Backward Euler’s method is more stable; that is, it will not be as affected by error over the long term. The Runge-Kutta method gives better approximations than forward Euler’s method because it has an order of 4. A method with order 4 implies that if the step size is cut in half, the new error will be , or the old error.