1.1.2.1. The Finite Difference Method¶

1.1.2.1.1. 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

1.1.2.1.2. 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

1.1.2.1.3. Define Derivative¶

1.1.2.1.3.1. Mathematical Interpretation¶

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

1.1.2.1.3.2. 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

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

1.1.2.1.3.3. Error¶

Some approximations are better than others
Quality of approximation improves as \(\Delta x\) is made smaller

1.1.2.1.3.4. 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 \(O(\Delta x)\) \(\rightarrow\) TRUNCATION ERROR
As \(\Delta x \rightarrow 0\) \(\Rightarrow\) FD converges!

Central Differencing¶

For Central Differencing, the error is \(O(\Delta x^2)\)