The truncation error in finite difference methods represents the discrepancy between the exact derivative and its approximation using a finite difference formula. This error arises from the inherent approximation involved in replacing a continuous differential operator with a discrete difference operator.
Understanding Discretization and Truncation Error
When solving partial differential equations (PDEs) numerically, we often use finite difference methods. This involves:
- Discretization: Approximating the continuous domain of the PDE with a grid of points.
- Difference Approximations: Replacing derivatives in the PDE with finite difference formulas.
The core issue is that difference approximations only approximate the actual differential operator. According to the reference:
The error between the numerical solution and the exact solution is determined by the error between a differential operator to a difference operator. This error is called the discretization error or truncation error.
This "error between a differential operator and a difference operator" is the truncation error.
How Truncation Error Occurs
Let’s consider a Taylor series expansion of a function f(x) around a point x:
f(x + h) = f(x) + hf'(x) + (h²/2!)f''(x) + (h³/3!)f'''(x) + ...
When we approximate a derivative (like f'(x)) using a finite difference formula, we essentially truncate this infinite series. For instance, the forward difference approximation of the first derivative f'(x) is:
f'(x) ≈ (f(x + h) - f(x)) / h
This approximation is obtained by taking the Taylor expansion f(x+h) and rearranging it to isolate f'(x), discarding the terms involving h² and higher. The neglected terms are the truncation error. In the above case, the truncation error is of the order O(h), which decreases as h gets smaller.
Key Aspects of Truncation Error
- Source: The approximation of derivatives by difference formulas.
- Nature: Error due to the truncation of an infinite series (like Taylor series).
- Impact: Influences the accuracy of the numerical solution; a smaller truncation error generally leads to a more accurate result.
- Dependence on Step Size: Typically decreases as the step size (h) becomes smaller, but this is not always the case when round-off errors become significant.
Practical Implications
The size of the truncation error depends on:
- Order of the Difference Approximation: Higher-order methods usually have smaller truncation errors but may require more computations.
- Step Size: Decreasing the step size typically decreases the truncation error, but there are diminishing returns and potential problems due to roundoff error.
Examples
- Forward Difference: The forward difference approximation for the first derivative has a truncation error of O(h), which decreases proportionally to h.
- Central Difference: The central difference approximation is more accurate and has a truncation error of O(h²), meaning it decreases proportionally to h². This shows how selecting a particular difference formula will influence the level of truncation error present.
Summary of Truncation Error in Finite Difference
| Feature | Description |
|---|---|
| Definition | Error from approximating a continuous derivative with a discrete difference operator. |
| Cause | Truncation of the infinite series used to derive finite difference formulas. |
| Impact | Influences the accuracy of the numerical solutions of differential equations. |
| Dependency | Dependent on step size h and the order of the difference approximation. Smaller h usually leads to less error but at a cost of computations. |
The truncation error is a crucial consideration when using finite difference methods for solving PDEs. Understanding it allows you to choose suitable approximations and optimize solutions for better accuracy.
