Using the current state and control variables of the airplane, the flight simulator calculates, by finite differences, the state of the airplane a small time increment in the future.

The first task for the time step calculation is to calculate variables dependent on the environment, including the air density and wind velocity; perhaps using Equations 2-4 to determine density.

The second task is to calculate the variables that depend on the state
and control variables. The simulator calculates the cosine matrix
using Equation 16 or Equation 21, depending
on how it stores orientation. If the simulator stores orientation
with quarternions, it then calculates the Euler angles using
Equations 30-32. It transforms the wind
velocity to body axes using Equation 7, and then
subtracts this from the aircraft velocity to yield the airspeed
components (
*u*_{a},*v*_{a},*w*_{a}). If it is important, the simulator may use
the formulas from Reference 3 to calculate the
geodetic coordinates (
).

The flight simulator calculates
and
using
Equations 5 and 6. It calculates
nondimensional parameters like
using
Equations 51-53. To calculate
,
it uses the chain rule with
and :

Equation 99 assumes and .

Once the necessary variables are calculated, the flight simulator uses the aerodynamic, landing gear, engine, and body force models from Sections 2, 3, and 4 to calculate the total resultant force and moment on the airplane.

Then it employs the the rigid body equations of motion (Equations 10-15, 17-19 or 22-25, and 33-35 or 36-38) to calculate the time derivatives of all the rigid-body state variables. Additionally, it uses Equations 65, 72, and 76, to calculate the time derivative of other state variables within the airplane.

Having determined the time derivatives of all state variables, the simulator is ready to use a finite-difference scheme to calculate the future state of the airplane. Generally, schemes that require only one calculation of the time derivatives per time step are preferred.

Reference 21 recommends using the predictive
Adams-Brashford method. For a given state variable *s*, the
method is given by Equation 100, in which
is the step size.

The method is second-order accurate. Another second order accurate method from Reference 21 is the trapezoidal method, given by Equation 101.

Reference 21 reports that both methods have acceptably low phase errors. Note that both methods require the time derivatives from the previous time step.