Local Error

Definition

Local error refers to the error made in a single step of a numerical method when approximating the solution of a differential equation. This type of error is crucial because it helps determine how accurate a numerical method is at each point in the solution process. Understanding local error is key for analyzing stability and convergence, as well as for comparing different numerical methods.

5 Must Know Facts For Your Next Test

1. Local error is usually expressed as a function of the step size, meaning that as you decrease the step size, local error typically decreases.
2. In many numerical methods, local error can often be derived analytically, allowing for better understanding of how each step contributes to the overall accuracy.
3. For explicit methods, local error can be more pronounced due to potential instability issues compared to implicit methods.
4. The order of local error is an important consideration; it tells you how fast the local error decreases as you refine your numerical approximation.
5. Many numerical algorithms, such as Runge-Kutta methods, have well-defined local errors that inform their performance and guide adjustments to improve accuracy.

Review Questions

• How does local error impact the selection of numerical methods for solving differential equations?
  • Local error plays a significant role in selecting numerical methods because it directly affects the accuracy of solutions at each step. Methods with lower local errors are generally preferred as they ensure that each step contributes less deviation from the true solution. This is particularly important for long-term integration where small errors can accumulate, leading to large global errors.
• Compare and contrast local error and global error in the context of numerical methods.
  • Local error is concerned with the accuracy of each individual step in a numerical method, while global error measures the total deviation of the approximate solution from the true solution after multiple steps. Understanding both types of errors is crucial; local errors contribute to global errors, and techniques to reduce local errors can help manage overall accuracy. The relationship between them helps identify whether issues stem from individual steps or accumulate over time.
• Evaluate how understanding local error can influence the design and implementation of numerical methods for stiff differential equations.
  • Understanding local error is vital when designing numerical methods for stiff differential equations because these problems often require specific approaches to handle rapid changes in solution behavior. For instance, implicit methods may be employed to maintain stability despite larger local errors during certain steps. By analyzing local error, developers can tweak parameters such as step size or select more suitable algorithms that manage stiffness efficiently while minimizing both local and global errors.

Related terms

Global Error: The cumulative error over all steps in a numerical method, which is the total deviation of the approximate solution from the true solution.

Consistent Method: A numerical method that approaches the exact solution as the step size decreases, meaning its local error converges to zero.

Stability: A property of a numerical method that indicates how errors are controlled and how they propagate through successive iterations.