Local truncation error

Definition

Local truncation error is the error introduced in a numerical method during a single step of the approximation process. This error arises when the exact solution is approximated by the numerical method, typically due to the finite difference used in approximating derivatives or integrating functions. It’s crucial to understand this error, as it helps evaluate the accuracy of methods like Euler's method and various multistep techniques, allowing us to refine these methods and achieve more precise results.

5 Must Know Facts For Your Next Test

1. Local truncation error for Euler's method is proportional to the square of the step size, meaning smaller steps can significantly reduce this error.
2. In multistep methods, understanding local truncation error helps in determining how many previous points to consider for improving accuracy.
3. Each numerical method has a specific formula to calculate its local truncation error, usually related to its order of accuracy.
4. Analyzing local truncation error allows for better adjustments to be made when selecting appropriate step sizes for achieving desired accuracy.
5. Reducing local truncation error often requires balancing between computational efficiency and the precision of the numerical solution.

Review Questions

• How does local truncation error affect the choice of step size in Euler's method?
  • Local truncation error directly influences the choice of step size in Euler's method because it indicates how much error is introduced with each step. A smaller step size results in a reduced local truncation error, which leads to more accurate solutions. However, smaller step sizes require more computations, so finding an optimal balance between accuracy and efficiency is essential.
• Compare and contrast local and global truncation errors in the context of numerical methods.
  • Local truncation error measures the error introduced during a single step of a numerical method, while global truncation error accounts for the accumulated errors over all steps. Understanding both types of errors is vital because local errors can accumulate into larger global errors. This distinction helps in evaluating how well a numerical method performs overall, guiding adjustments to minimize both types of errors.
• Evaluate the impact of local truncation error on the accuracy of multistep methods compared to single-step methods.
  • Local truncation error plays a critical role in determining the accuracy of multistep methods as it affects how past computed values influence current approximations. Multistep methods often provide better accuracy for larger intervals due to reduced local truncation errors across steps, leveraging previous values effectively. However, if not managed correctly, they can also compound errors more than single-step methods, making it essential to analyze and optimize both local and global truncation errors for reliable results.

Related terms

global truncation error: The cumulative error over all steps in a numerical method, which is influenced by the local truncation error at each step.

order of accuracy: A measure of how the error decreases as the step size is reduced in a numerical method, typically described as first-order or second-order.

step size: The increment in the independent variable used in numerical methods, affecting both local and global truncation errors.