Numerical Starting Procedure

At time $t=0$ , the displacement $u$ and velocity ${\dot{u}}_0$ are known from initial conditions. The acceleration ${\ddot{u}}_0$ and time step $h_1$ are found from solving the equations of motion. The initial time step $h_0$ is set to zero:(1)

$$h_0=0;h_{\frac{1}{2}}=\frac{h_1}{2}$$

(2)

$${\dot{u}}_{-\frac{1}{2}}={\dot{u}}_0$$

(3)

$${\dot{u}}_{\frac{1}{2}}={\dot{u}}_0+h_{\frac{1}{2}}{\ddot{u}}_0$$