Global Error

Definition

Global error refers to the total difference between the exact solution of a differential equation and the numerical approximation produced by a numerical scheme over the entire domain of interest. This concept is crucial when assessing the accuracy of numerical methods, especially in relation to their stability and consistency, as it helps to determine how closely a numerical solution approximates the true solution over time or space.

5 Must Know Facts For Your Next Test

1. Global error is typically affected by both local errors and the propagation of these errors through multiple steps in a numerical method.
2. In finite difference methods, global error can accumulate due to inaccuracies at each discrete point in time or space, impacting the overall solution.
3. For stable and consistent methods, the global error generally decreases as the mesh size is refined, leading to better approximations of the exact solution.
4. The relationship between global error and convergence is critical; if a numerical scheme is not converging, the global error will not diminish even as local errors improve.
5. Analyzing global error helps identify whether a numerical scheme is effective for specific types of differential equations, particularly parabolic and hyperbolic equations.

Review Questions

• How does global error relate to local error in numerical schemes?
  • Global error is the accumulation of local errors that arise from each discrete step in a numerical method. While local error refers to inaccuracies at individual steps, global error measures the total deviation from the exact solution across the entire computational domain. Understanding this relationship is essential because if local errors are small but not properly controlled, they can lead to significant global errors over many iterations.
• Discuss how stability and consistency influence global error in numerical methods.
  • Stability and consistency are vital factors that influence global error in numerical methods. A stable method ensures that small changes in initial conditions or inputs do not result in large fluctuations in the output, helping to maintain a manageable global error. Consistency guarantees that as the mesh size decreases, local errors converge towards zero. Together, these properties imply that if a method is both stable and consistent, it can produce a bounded global error that decreases with finer discretizations.
• Evaluate how understanding global error can impact the choice of numerical methods for solving differential equations.
  • Understanding global error is crucial when selecting numerical methods for solving differential equations because it provides insights into how well different methods will approximate solutions under various conditions. Evaluating global error allows for comparisons between methods based on their performance over time or space, especially for complex equations. This evaluation helps inform decisions on which method to apply based on accuracy requirements and computational resources available, ultimately leading to more effective problem-solving strategies.

Related terms

Local Error: Local error is the error introduced in a single time step or spatial step of a numerical scheme. It is typically dependent on the method used and the size of the discretization.

Convergence:

Convergence refers to the property of a numerical method where the approximate solution approaches the exact solution as the discretization parameters, such as step size, tend to zero.

Stability: Stability describes how small changes or perturbations in the initial conditions or numerical parameters affect the behavior of a numerical solution over time.