Global error

Definition

Global error refers to the total accumulated difference between the exact solution of a mathematical problem and the approximate solution obtained through numerical methods. It reflects how far off a numerical approximation is from the true solution over the entire domain, rather than just at specific points. This measure is crucial when evaluating the accuracy and reliability of numerical methods, particularly in finite difference methods, where approximations are used to solve differential equations.

5 Must Know Facts For Your Next Test

1. Global error can be affected by both local error and the number of steps taken in a numerical method; more steps can lead to better approximations if local errors are minimized.
2. In finite difference methods, global error is often analyzed to determine the overall effectiveness of the method in solving partial differential equations.
3. A key factor in minimizing global error is choosing an appropriate grid size; smaller grid sizes typically yield more accurate results but require more computational resources.
4. The relationship between global error and step size can often be described mathematically, helping to establish the convergence rate of a numerical scheme.
5. Different numerical methods can exhibit varying levels of global error based on their formulation and implementation, making it essential to choose the right approach for specific problems.

Review Questions

• How does global error differ from local error in numerical methods?
  • Global error encompasses the total discrepancy between the exact and approximate solutions over the entire domain, while local error focuses on discrepancies at individual steps. Local errors can accumulate across multiple steps, contributing to the global error. Understanding both types of errors is vital for assessing the overall accuracy of numerical methods.
• Discuss how grid size impacts global error in finite difference methods.
  • Grid size plays a significant role in determining global error in finite difference methods. A finer grid typically results in lower local errors at each point, which helps reduce global error. However, decreasing grid size increases computational costs. Therefore, striking a balance between accuracy and efficiency is crucial when selecting grid sizes to minimize global error.
• Evaluate how convergence affects global error in numerical solutions, particularly in finite difference methods.
  • Convergence has a direct impact on global error as it indicates how closely a numerical solution approaches the exact solution as step sizes decrease. In finite difference methods, if a method converges properly, reducing step sizes should lead to a significant decrease in global error. This relationship helps practitioners understand how modifications to their numerical approach can enhance accuracy and reliability when solving complex problems.

Related terms

local error: Local error measures the error introduced at a single step of a numerical method, indicating how accurate the approximation is at that particular point.

convergence: Convergence describes the behavior of a numerical method as it approaches the exact solution as the step size decreases, ideally leading to a reduction in global error.

stability: Stability refers to the property of a numerical method that ensures small perturbations or errors in initial conditions do not lead to large deviations in the computed solution.