Graphical Explanation of Euler and Modified Euler Methods

Figure 1: Graphical representation of the Euler and modified Euler method. See text.

Figure 1 illustrates the method. The exact solution curve passes through point A at time $t$ on its way to point D at time $t+h$. We would like to step from A to D. The simple Euler method uses the ODE to evaluate the slope of the tangent at A. It then steps along the tangent to point B, which represents the Euler estimate for D. A different exact solution curve passes through point B as shown, obviously not the same as the exact solution curve passing through A and D. The problem here is that the solution curve has a large second derivative, so the local truncation error is large. The slope at A is too steep. A smaller slope would have been better.

For the modified Euler method, point B is a provisional point. The modified Euler method evaluates the slope of the tangent at B, as shown, and averages it with the slope of the tangent at A to determine the slope of the improved step. Averaging is an improvement because the slope at B is too shallow while the slope at A is too steep. The line AC represents the modified Euler step with the average slope, leading to the improved approximation C. Of course, it is still not perfect, and lies on a slightly different solution curve from the one passing through A and D.