finite_difference_method.rst

The Finite Difference Method

Origin and Concept

• Invented by Euler in 1768 for one dimension, extended by Runge in 1908 to two dimensions
• Concept is to approximate derivatives using Taylor Expansions

Define numerical grid

• Two families of grid lines
• Grid lines of the same family do not intersect
• Grid lines of different families intersect only once
• Node (i,j) is in 2D - an unknown field variable which depends on neighbouring nodes providing one algebraic equation

Define Derivative

Mathematical Interpretation

$$\left . {\partial u \over \partial x} \right \vert_i = \lim_{\Delta x \rightarrow 0} {u(x_i + \Delta x) - u(x_i) \over \Delta x}$$

Geometric Interpretation

• Slope of the tangent to the curve with three approximation to the exact solution: Backward, Forward and Central Difference.

• Backward difference

$$\left . {\partial u \over \partial x} \right \vert_i \approx {{u_i - u_{i-1}} \over \Delta x}$$

• Forward difference

$$\left . {\partial u \over \partial x} \right \vert_i \approx {{u_{i+1} - u_i} \over \Delta x}$$

• Central difference

$$\left . {\partial u \over \partial x} \right \vert_i \approx {{u_{i+1} - u_{i-1}} \over 2 \Delta x}$$

Error

• Some approximations are better than others
• Quality of approximation improves as Δx is made smaller

Taylor Series Expansion - Order of the approximations

$$u(x) = u(x_i)+(x-x_i) \left . {\partial u \over \partial x} \right \vert_i + {(x - x_i)^2 \over 2!} \left . {\partial^2 u \over \partial x^2} \right \vert_i + \cdots + {(x - x_i)^n \over n!} \left . {\partial^n u \over \partial x^n} \right \vert_i$$

• Forward differencing: x = x_i + 1
• Backward differencing: x = x_i − 1

Forward Differencing

• We need to obtain the derivative $\left . {\partial u \over \partial x} \right \vert_i$

$$\left . {\partial u \over \partial x} \right \vert_i = {(u_{i+1} - u_i) \over (x_{i+1} - x_i)} - {(x_{i+1} - x_i) \over 2!} \left . {\partial^2 u \over \partial x^2} \right \vert_i - \cdots - {(x_{i+1} - x_i)^{n-1} \over n!} \left . {\partial^n u \over \partial x^n} \right \vert_i$$

• If x_i + 1 − x_i is small, then:

$$\left . {\partial u \over \partial x} \right \vert_i = {(u_{i+1} - u_i) \over \Delta x} - O(\Delta x)$$

• There is a possibility that the derivative $\left . {\partial^2 u \over \partial x^2} \right \vert_i$ is large, but we assume that the function is well-behaved.
• Forward differencing approximation neglected terms of O(Δx) → TRUNCATION ERROR
• As Δx → 0 ⇒ FD converges!

Central Differencing

• For Central Differencing, the error is O(Δx²)