ODE Approximation

Euler Forward Method

Euler's forward method is

$$f_n = \frac{y_{n+1} - y_n}{\Delta t}$$

With a forward approximation, we want to look ahead the next value with the current value known. So rearrange the equation to isolate the current values on the right

$$y_{n+1} = y_n + f_n \Delta t$$

$y_n$ is known and $\Delta t$ is chosen by us. What is $f_n$? It's the function that we want to approximate.

$$\frac{dy}{dt} = f(t_n, y_n)$$

Since we know $y_n$ (it's the current value) and $t_n$ ( the current time is the number of steps $n$ times the step size $\Delta t$ ), simply feed them to our ODE to find the $f_n$ required.

Here's an example with $\frac{dy}{dt} = 2y$

$$y_{n+1} = y_n + 2 y_n \Delta t \\ y_{n+1} = y_n ( 1 + 2 \Delta t )$$

Now the next value is always computable given the current value and a step size. However, what we had was always acceptable as a template for writing some code.

def forward_step(equation, step_size, step_number, current_value):
    t = step_size * step_number
    return current_value + equation(current_value, t) * step_size

def forward_approximation(equation, step_size, number_of_steps, starting_value):
    current_value = starting_value
    for step_number in range(0, number_of_steps):
        current_value = forward_step(equation, step_size, step_number, current_value)
    
    return current_value

This is not good code, however, as the error from the true value is unacceptable and the runtime is inefficient. But Euler's method will get you thinking about approximation functions, and for that, it's useful to know.

See a longer write up here