Global error

Definition

Global error refers to the cumulative difference between the exact solution of a differential equation and the approximate solution obtained through numerical methods over the entire interval of interest. This type of error encompasses all the local errors that occur at each step of the numerical integration process, making it a critical factor in assessing the accuracy and stability of multistep methods used for solving ordinary differential equations.

5 Must Know Facts For Your Next Test

1. Global error is typically influenced by both local errors and the number of steps taken in a numerical method; more steps can either reduce or exacerbate global error depending on stability.
2. In multistep methods, global error can be analyzed using Taylor series expansions to understand how truncation affects overall accuracy.
3. Global error is crucial for determining whether a numerical method is suitable for solving stiff equations, as high global error can lead to incorrect solutions.
4. Different multistep methods have different orders of accuracy, which directly affects how quickly global error decreases as the step size becomes smaller.
5. Controlling global error is essential for ensuring that numerical solutions remain valid and reliable over long intervals, especially when simulating dynamic systems.

Review Questions

• How does global error relate to local error in numerical methods, particularly in multistep approaches?
  • Global error is derived from the accumulation of local errors that occur at each step when using numerical methods like multistep approaches. While local error focuses on mistakes made in a single step, global error reflects how these mistakes compound across all steps. Understanding this relationship helps in evaluating the overall accuracy and effectiveness of numerical methods.
• Discuss how stability influences global error in multistep methods and its implications for solving differential equations.
  • Stability plays a crucial role in managing global error within multistep methods. If a method is stable, small perturbations will not lead to significant increases in global error, thereby maintaining accuracy. Conversely, an unstable method can cause global error to grow rapidly, resulting in unreliable solutions. Therefore, selecting a stable method is essential when solving differential equations numerically.
• Evaluate the impact of convergence on global error for multistep methods when applied to stiff differential equations.
  • Convergence significantly affects global error in multistep methods, especially for stiff differential equations where traditional methods may struggle. A method that converges quickly will typically exhibit reduced global error, ensuring that solutions are accurate even over extended intervals. Conversely, if convergence is slow or problematic, global error may remain high, leading to invalid results. Thus, understanding convergence properties is vital for selecting appropriate numerical techniques for challenging differential equations.

Related terms

local error:

Local error is the error made in a single step of a numerical method, representing the difference between the exact value and the approximate value after one step.

stability:

Stability in numerical methods refers to the property that small changes in the initial conditions or inputs lead to small changes in the outputs, ensuring that errors do not grow uncontrollably during computations.

convergence: Convergence is the property of a numerical method that describes how closely the approximate solution approaches the exact solution as the number of steps increases or as the step size decreases.