# Derive local truncation error for the Improved Euler Method

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In summary, the local truncation error for the autonomous ODE is |x(t1) − x1| and can be found using the Improved Euler method. By expanding the right-hand side of the method's second equation and comparing it to the Taylor series of the correct value, the error can be determined to be O(h^3). It is also helpful to search for the local truncation error for the Improved Euler method for more information.
oceanix
TL;DR Summary
Can't figure out how to find the local truncation error for the Improved Euler method
I'm trying to find the local truncation error of the autonomous ODE: fx/ft = f(x).

I know that the error is |x(t1) − x1|, but I can't successfully figure out the Taylor expansion to get to the answer, which I believe is O(h^3).

Any help would be greatly appreciated!

Did you try googling " local truncation error for the Improved Euler " ? I got lots of good hits.

Is that \begin{align*} \bar x_{i+1} &= x_i + hf(x_i) \\ x_{i+1} &= x_i + \tfrac12 h(f(x_i) + f(\bar x_i))\mbox{?}\end{align*} The method is to expand the right-hand side of the second equation in a Taylor series about $x_i$,
\begin{align*} x_{i+1} &= x_i + \tfrac12 h \left(f(x_i) + f(x_i + hf(x_i))\right) \\ &= x_i + \tfrac12 h \left(2f(x_i) + hf(x_i)f'(x_i) + \tfrac12h^2f(x_i)^2f''(x_i) + O(h^3) \right) \end{align*}
and compare it to the Taylor series of the correct value, \begin{align*} x(t_i + h) &= x(t_i) + h\dot x(t_i) + \tfrac12 h^2 \ddot x(t_i) + \tfrac16 h^3 x^{(3)}(t_i) + O(h^4) \\ &= x_i + hf(x_i) + \tfrac12h^2 f'(x_i)f(x_i) + \tfrac16 h^3 f(x_i)(f(x_i)f''(x_i) + f'(x_i)^2) + O(h^4). \end{align*}

## 1. What is the Improved Euler Method?

The Improved Euler Method is a numerical method used to approximate the solution to a differential equation. It is an extension of the Euler Method and is also known as the Heun's Method or the Modified Euler Method.

## 2. How does the Improved Euler Method work?

The Improved Euler Method uses a two-step process to approximate the solution to a differential equation. In the first step, it uses the Euler Method to estimate the value of the solution at the midpoint of the interval. In the second step, it uses this midpoint value to calculate a new estimate for the solution at the end of the interval.

## 3. What is local truncation error?

Local truncation error is the difference between the exact solution of a differential equation at a given point and the approximate solution obtained by a numerical method at the same point. It is a measure of the accuracy of the numerical method and is typically expressed as a function of the step size used in the method.

## 4. How do you derive the local truncation error for the Improved Euler Method?

The local truncation error for the Improved Euler Method can be derived by comparing the exact solution of the differential equation with the two-step approximation obtained by the method. This involves using Taylor series expansion and then taking the difference between the two expressions. The resulting expression gives the local truncation error as a function of the step size and the derivative of the solution.

## 5. How can the local truncation error be reduced in the Improved Euler Method?

The local truncation error in the Improved Euler Method can be reduced by decreasing the step size used in the method. This means taking smaller intervals and using more steps to approximate the solution. Additionally, using a higher-order numerical method, such as the Runge-Kutta method, can also reduce the local truncation error.

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