Outlook

• 1D Finite difference schemes
  • First derivative
  • Example 1: Finding maxima/minima as roots
  • General difference schemes
  • Example 2: Transmission through
  • Irregular difference schemes
• Ordinary differential equations
  • A simple ordinary differential equation

1D Finite difference schemes

• Formulas for the derivative \(f_i' \equiv f'(x_i)\) at \(x_i\) are:

Best scheme?

• For on-the-fly calculations, only “old” values available: use backward scheme
• At boundaries: forward for left, backward for right boundary
• Other cases: central difference often has a smaller error

Best scheme?

• For on-the-fly calculations, only “old” values available: use backward scheme
• At boundaries: forward for left, backward for right boundary
• Other cases: central difference often has a smaller error \[ f_{i, \textrm{central}}' = f_i' + f_{i}'''\frac{h^2}{6} + \mathcal{O(h^4)}\quad \Longrightarrow \quad E_\textrm{central} \approx f_{i}'''\frac{h^2}{6} \] \[ f_{i, \textrm{forward}}' = f_i' + f_{i}''\frac{h}{2} + \mathcal{O(h^2)}\quad \Longrightarrow \quad E_\textrm{forward} \approx f_{i}''\frac{h}{2} \]

Example error for sine function

• Example: \(f(x) = \sin(x)\), \(f' = \cos(x)\), \(f'' = -\sin(x)\)
  • Error central difference mostly smaller than forward difference
  • Central(forward) difference performs bad(good) at inflection points of the function (curvature flips sign)

Calculation of the error

• The formula for the central difference: \[ f'_{i, \textrm{central}} = \frac{f_{i+1} - f_{i-1}}{2h} \]
• Taylor expand the function \(f_{i \pm 1}\)

\[ f_{i \pm 1} = f_{i} \pm f_{i}'h + f_{i}''\frac{h^2}{2!} \pm f_{i}'''\frac{h^3}{3!} + f_{i}''''\frac{h^4}{4!} \pm \dots \]

\(\longrightarrow\) Use the Taylor expansion in the central difference formula

Calculation of the error

Numerator of \(\,\, f_{i,\textrm{central}}'\):

\[ f_{i+1} - f_{i-1} = \left[f_{i} + f_{i}'h + f_{i}''\frac{h^2}{2!} + f_{i}'''\frac{h^3}{3!} + f_{i}''''\frac{h^4}{4!}\right]\\ - \left[f_{i} - f_{i}'h + f_{i}''\frac{h^2}{2!} - f_{i}'''\frac{h^3}{3!} + f_{i}''''\frac{h^4}{4!}\right] \]

\[ \Longrightarrow \qquad \frac{f_{i+1} - f_{i-1}}{2h} = \frac{2f_{i}' h + 2 f_{i}'''\frac{h^3}{3!} + \mathcal{O(h^5)}}{2h} \]

\[ f'_{i, \textrm{central}} = f_{i}' + f_{i}'''\frac{h^2}{3!} + \mathcal{O(h^4)} \]

Calculation of the error

\[ f'_{i, \textrm{central}} = f_{i}' + f_{i}'''\frac{h^2}{3!} + \mathcal{O(h^4)} \]

The error \(E\) is the approx. derivative minus the real one:

\[ E \approx f'_{i, \textrm{central}} - f_{i}' = f_{i}'''\frac{h^2}{6} + \mathcal{O(h^4)} \]

Finite differences in matlab

• Defining an expression/function in Matlab
• Function handles in Matlab: fh = @exp
• Anonymous function handles: fh = @(x, y, a) x.^2 + a*y

x = linspace(0, 1, 11);
f = fh(x, 2);
fh = @(x, a) exp(-a*x^2) * sin(3*x); 

fx = diff(f) / diff(x)    % Forward/backward difference
F = -X.^2 + Y.^2;         % Example 2D Scalar field
[Fx, Fy] = gradient(F)    % Gradient
laplacian = del2(F)       % Laplacian

Finding minima/maxima

• Minima/maxima at roots of the derivative
• Use root finding

x = linspace(0, 1, 11);
f = fh(x, 2);
fh = @(x, a) exp(-a*x^2) * sin(3*x); 

General finite differences

• A general formula for higher derivatives:

\[ \frac{d^n f_j}{dx^n} = \frac{1}{h^n} \sum_{l=a}^b c_l f_{j+l} \]

• Finding the coefficients \(c_l\): Taylor expansion

\[ f_{i \pm l} = f_{i} \pm f_{i}'\,lh + f_{i}''\frac{l^2h^2}{2!} \pm f_{i}'''\frac{l^3h^3}{3!} + f_{i}''''\frac{l^4h^4}{4!} \pm \dots \]

• Fill in Taylor expansion into the formula

General finite differences

\[ \frac{d^n f_j}{dx^n}\left|_\textrm{num}\right. = \frac{1}{h^n} \sum_{l=a}^b c_l f_{j+l} = \sum_{m=0}^\infty \frac{1}{m!} h^{m - n} \left( \sum_{l=a}^b \right) \frac{d^m f_j}{dx^m} \]

terms with \(m < n\) are zero:

\[ \textrm{For }m = 0 \dots n-1, \qquad \sum_{l = a}^b l^m c_l = 0 \]

term with \(m = n\) should be the derivative:

\[ \frac{1}{n!} \sum_{l=a}^b l^n \,c_l = 1 \]