Guide to Non-Dimensionalization

The speed of a light is a large number within our calculations. As such, when it is used in a calculation in a computer, numbers being add to the speed of light that are much smaller will not be recognized. For example, if I want to calculate $10^8+1$, it will result in $10^8$ instead of the more accurate number. Furthermore, if a number grows as large as $10^{16}$, the computer will register it as $1$. Both of these errors come from large formulas being calculated in the computer. Therefore, to reduce these errors, Nondimensionalization is needed.

Nondimensionalization is a technique used in ordinary differential equations that partially or completely physical dimensions from equations that have physical qualities by substituting variables. This is done by doing the following:

1. Identify all independent and dependent variables
2. Replace each of the variables with a new variable that is scaled by a characteristic unit
3. Divide the equation by the coefficient of the highest order polynomial or derivative term
4. Define the characteristic units such that the coefficients of each ordered term equals 1.
5. Rewrite the system of equations in terms of their new dimensionless quantities

For example, consider a first order differential equation with constant coefficients:

$$\begin{array}{c}a\frac{dx}{dt}+bx=Af(t)\end{array}$$

Here, $t$ is an independent variable and $x$ is a dependent variable. Therefore, let $t=\tau t_c$ and $x=\chi x_c$ where $t_c$ and $x_c$ are the characteristic units and $\tau$ and $\chi$ are the new variables. Then,

$$\begin{array}{rl}& a\frac{dx}{dt}+bx=Af(t)\\ & =>a\frac{d}{d\tau t_c}(\chi x_c)+b\chi x_c=Af(\tau t_c)\\ & =>a\frac{x_c}{t_c}\frac{d\chi }{d\tau }+b\chi x_c=Af(\tau )\end{array}$$

Diving the equation of by the coefficients of the higheest order derivative term, $a\frac{x_c}{t_c}$,

$$\begin{array}{c}\frac{d\chi }{d\tau }+\frac{b{t}_c}{a}\chi =\frac{A{t}_c}{a{x}_c}f(\tau )\end{array}$$

Defining the characteristic units such that as many cofficients of each term become $1$,

$$\begin{array}{rl}& \frac{b{t}_c}{a}=1\\ & =>t_c=\frac{a}{b}\end{array}$$ $$\begin{array}{rl}& \frac{A{t}_c}{a{x}_c}=1\\ & =>\frac{1}{x_c}=\frac{a}{A{t}_c}\\ & =>x_c=\frac{A{t}_c}{a}=\frac{A}{b}\end{array}$$

Rewriting the equation in the terms of the characteristic units,

$$\begin{array}{c}\frac{d\chi }{dt}+\chi =f(\tau )\end{array}$$