Spring Mass Systems:  Midpoint Method for Solving ODEs

If the system is stiff, i.e., if the constant k is large,
then the time steps will have to be rather small to yield a truthful simulation.
Fortunately there are better methods to solve differential equations.
Here is a first simple improvement:

Midpoint Method

a. Compute an Euler step: Dx = Dt f(x, t) b. Evaluate f at the midpoint: f mid = f( (x + Dx)/2 ,  ( t + Dt)/2 ) c. Take a step using the midpoint value: x(t + Dt) = x(t) + Dt fmid

A generalization of this method using higher-order terms in the Taylor series leads to the Runge-Kutta method.

To learn more:

Physically Based Modeling: Principles and Practice -- Particle System Dynamics"  by  Andrew Witkin