29–32. {Use of Tech} Errors in Euler’s method Consider the following initial value problems.


d. In general, how does halving the time step affect the error at t=0.2 and t=0.4?


y′(t) = y/2, y(0) = 2; y(t) = 2eᵗᐟ²

Verified step by step guidance

1. Recall that Euler's method approximates the solution to the differential equation by using the formula \(y_{n+1} = y_n + h f(t_n, y_n)\), where \(h\) is the time step size.
2. Understand that the local truncation error of Euler's method at each step is proportional to \(h^2\), and the global error (error accumulated over multiple steps) is proportional to \(h\).
3. Since the global error is roughly proportional to the step size \(h\), halving the time step \(h\) will approximately halve the error at any fixed time \(t\), such as \(t=0.2\) and \(t=0.4\).
4. To see this concretely, consider the number of steps needed to reach \(t=0.2\) and \(t=0.4\) with step size \(h\) and with step size \(h/2\). The smaller step size means more steps but smaller error per step, resulting in an overall smaller total error.
5. Therefore, halving the time step reduces the global error roughly by a factor of 2 at the specified times, improving the accuracy of Euler's method for this problem.

Key Concepts

Euler's Method

Euler's method is a numerical technique to approximate solutions of initial value problems for ordinary differential equations. It uses a stepwise approach, estimating the next value by moving along the slope (derivative) at the current point multiplied by a small time step. The accuracy depends on the step size chosen.

Local and Global Truncation Error

Local truncation error is the error made in a single step of Euler's method, while global truncation error accumulates over multiple steps. For Euler's method, the local error is proportional to the square of the step size, and the global error is proportional to the step size, meaning smaller steps reduce overall error.

Effect of Step Size on Error

Halving the time step in Euler's method generally reduces the global error approximately by half, improving accuracy. This is because the global error is linearly dependent on the step size, so smaller steps lead to more precise approximations at given points like t=0.2 and t=0.4.