Euler's Method Error: Derivation & Estimation

[Moose352]

I understand how the (local) error for euler's method of integration is derived from the perspective of the taylor expansion and inequality. However, I don't really see why taylor's equation needs to be invoked, since the euler method can also be derived as a tangent line approximation. How then is the order of the error estimated by interpreting euler's method as a tangent line approximation?

 

[Integral]

It is really the same thing. Using a Taylor series expansion, for the Euler method you truncate all but the linear terms, thus your approximation is assuming a local linear function. This is the exact same thing as using a tangent line approximation. You are assuming that the function is locally linear. When you say you are just making a tangent line approximation you are simply ignoring the fact that you are in reality just dropping the nonlinear terms of the Taylor series.

 

[tacman]

Euler's method is a numerical method for approximating the solutions of differential equations. It is based on the idea of using tangent lines to approximate the curve of the solution at discrete points. This method has a local error, which refers to the difference between the exact solution and the approximate solution at a single point. The local error is important because it can affect the accuracy of the overall approximation.

To understand the derivation and estimation of the error in Euler's method, let's first consider the method from a geometric perspective. As mentioned, the method uses tangent lines to approximate the curve of the solution. These tangent lines are calculated using the slope of the differential equation at a given point. However, since the differential equation is a continuous function, the slope at a single point is not enough to accurately determine the slope of the tangent line. This is where Taylor's expansion comes into play.

Taylor's expansion is a mathematical tool that allows us to approximate a function using its derivatives at a given point. By using higher-order derivatives, we can get a more accurate approximation of the function. In the case of Euler's method, we use Taylor's expansion to approximate the slope of the tangent line by taking into account the second derivative of the function. This is necessary because the first derivative alone may not provide a good enough approximation for the tangent line.

Now, let's move on to the estimation of the error in Euler's method. As mentioned earlier, the local error is the difference between the exact solution and the approximate solution at a single point. This error can be estimated by looking at the difference between the exact tangent line and the approximate tangent line. This difference is affected by the second derivative of the function, which is why Taylor's expansion is necessary.

By using higher-order derivatives in the Taylor's expansion, we can estimate the order of the error in Euler's method. The order of the error refers to how the error changes as we decrease the step size (h) of the method. A higher order of the error means that the error decreases at a faster rate as h decreases, resulting in a more accurate approximation.

In conclusion, while it is possible to derive Euler's method as a tangent line approximation, the use of Taylor's expansion is necessary to accurately estimate the error of the method. It allows us to account for the curvature of the function and provides a more accurate estimation of the error.