ODE solvers: Euler vs RK4 vs adaptive RK45

What you are seeing: all three methods integrate $\ddot x = -\omega^2 x$ from the same initial condition. Euler's explicit step pumps energy upward without bound; RK4 stays close to the true orbit at fourth-order accuracy; adaptive RK45 chooses its own step size from local error estimates. The phase-space plot at right shows each trajectory in $(x, v)$.