Numerical integration methods: Explicit vs Semi-Implicit vs Newton-Euler 1, 2 and use in cyclone physics engine

Question

I am trying to understand the difference between explicit Euler and semi-implicit Euler integration, where in explicit Euler the current position is calculated as

$$x_{n+1} = x_n + v_n$$

and semi-implicit Euler method where it is calculated as:

$$x_{n+1} = x_n + v_{n+1}$$

with $x$ being position and $v$ the velocity, $n$ is a time, $n+1$ the time at next update.

In particular, can anyone confirm, which of the two methods is used in this code

If I understand, it is semi-implicit Euler, correct? Because the velocity is updated, before being added to the position, right?

If my reasoning is wrong, please let me know how a semi-implicit update would look like then? Or why I am wrong?

Also, I am having trouble with terminology: I've read about an integration called Newton-Euler 1 and another called Newton-Euler 2. Where Newton-Euler 1 was described as less accurate, and only uses the first-order differentiation for calculating the new position, while Newton-Euler 2 also adds the 2nd order differentiation. I've seen the wiki entry for Newton Euler equations, but it doesn't call an equation "1" or "2". The book I've read this in is accompanied by the code given in the link above.

How (if at all) are Newton-Euler 1,2 related to explicit and semi-implicit Euler methods?

Answer 1

Looking at the code, unless there is something weird under the hood, it is semi-implicit Euler.

I do not know any integration/differentiation technique called Newton-Euler. From what I read, it is a formulation to describe motion and a competitor to Lagrangian formulation. The book might be talking about the particular forms first and second order explicit Euler method when applied to Newton-Euler description of motion.