Local truncation error

Definition

Local truncation error refers to the error introduced in a numerical method at a single step of computation, representing the difference between the exact solution and the numerical approximation after one step. This type of error is significant when evaluating the accuracy of numerical methods for solving ordinary and partial differential equations, as it provides insight into how errors accumulate through iterative processes.

5 Must Know Facts For Your Next Test

1. Local truncation error is primarily associated with the method used for approximating derivatives in differential equations.
2. This type of error is usually expressed as a function of the step size raised to a power, showing how it decreases as the step size gets smaller.
3. In many numerical methods, local truncation error is directly related to the order of the method; higher-order methods generally have lower local truncation errors.
4. Understanding local truncation error helps in selecting appropriate step sizes for numerical methods, balancing computational efficiency and accuracy.
5. Local truncation error can be analyzed using Taylor series expansion, where the remainder term indicates the degree of approximation.

Review Questions

• How does local truncation error affect the overall accuracy of numerical solutions in solving differential equations?
  • Local truncation error impacts overall accuracy by determining how much error is introduced at each individual step of a numerical method. If local errors are large, they can propagate through subsequent steps, leading to an increased global truncation error. Thus, understanding and minimizing local truncation error is crucial for obtaining reliable solutions, particularly when dealing with complex differential equations.
• Discuss the relationship between local truncation error and step size in numerical methods. How does changing the step size influence this type of error?
  • Local truncation error is influenced significantly by the choice of step size; generally, a smaller step size results in a lower local truncation error. As the step size decreases, the approximation becomes more accurate, reducing the difference between the exact solution and the computed value after each step. However, using too small a step size can lead to increased computational costs and potential round-off errors, making it essential to find a suitable balance.
• Evaluate how local truncation error can guide you in selecting a numerical method for solving a particular problem involving ordinary or partial differential equations.
  • When choosing a numerical method for solving differential equations, evaluating local truncation error provides insights into the method's accuracy and efficiency. Different methods exhibit varying orders of convergence; thus, understanding their local truncation errors allows you to select one that meets specific precision requirements without excessive computational effort. By assessing these errors alongside factors like stability and computational load, you can make informed decisions that enhance solution reliability while managing resource use effectively.

Related terms

global truncation error: The cumulative error that accumulates over all steps in a numerical method, reflecting the overall accuracy of the solution over the entire interval of interest.

numerical stability: A property of a numerical algorithm that indicates how errors are propagated during computation, ensuring that small changes in initial conditions or inputs do not lead to large variations in results.

step size: The increment used in numerical methods when progressing through the domain, which affects both local and global truncation errors; smaller step sizes typically reduce these errors.