๐ Understanding Local and Global Errors in Numerical ODEs
When solving Ordinary Differential Equations (ODEs) numerically, we're essentially approximating the solution at discrete points. Two important concepts in this context are local error and global error. Let's break them down and then compare them.
๐ Definition of Local Error
Local error, also called local truncation error, is the error introduced in a single step of the numerical method, assuming that the previous step's solution is exact. In other words, it's the difference between the numerical solution after one step ($y_{i+1}$) and the exact solution ($y(t_{i+1})$) of the ODE at time $t_{i+1}$, given that the numerical solution at the previous time step $t_i$ ($y_i$) is equal to the exact solution $y(t_i)$. Mathematically:
$Local Error = y_{i+1} - y(t_{i+1})$ given that $y_i = y(t_i)$
๐ Definition of Global Error
Global error, also known as accumulated error, is the difference between the numerical solution at a given time $t_i$ ($y_i$) and the exact solution $y(t_i)$ of the ODE at that same time. This error takes into account the accumulation of errors from all previous steps. Mathematically:
$Global Error = y_i - y(t_i)$
๐ Comparison Table: Local vs. Global Error
| Feature |
Local Error |
Global Error |
| Definition |
Error introduced in a single step, assuming the previous step is exact. |
Accumulation of errors over all previous steps. |
| Assumption |
Previous step's solution is exact. |
Previous steps' solutions may contain errors. |
| Accumulation |
No accumulation (single step). |
Accumulated over all steps. |
| Order of Magnitude |
Typically related to the step size ($h$) raised to a power, e.g., $O(h^{p+1})$ where $p$ is the order of the method. |
Typically one order lower than the local error, e.g., $O(h^p)$. |
| Practical Use |
Helps in analyzing the stability and consistency of a numerical method. |
Provides an overall measure of the accuracy of the solution at a given time. |
๐ Key Takeaways
- ๐ Scope: Local error focuses on a single step, while global error considers the entire solution.
- ๐ Accumulation: Global error accumulates errors from previous steps; local error does not.
- ๐ Magnitude: The global error is generally one order lower than the local error. Reducing the step size ($h$) decreases both errors, but the relationship isn't always linear due to error propagation.
- ๐งช Practical Implications: Understanding these errors helps in choosing appropriate numerical methods and step sizes to achieve desired accuracy.