Realtime aircraft dynamics simulation requires very high accuracy and stability in the numerical integration process. Nonetheless, traditional multistep numerical methods cannot effectively meet the new requirements. Therefore, a novel realtime multistep method based on PredictEvaluateCorrect scheme of threestep fourthorder method (RTPEC34) is proposed and developed in this research to address the gap. In addition to the development of a highly accurate algorithm based on predictorcorrector, the contribution of this work also includes the analysis of truncation error for realtime problems. Moreover, the parameters for the RTPEC34 method are optimized using intelligent optimization algorithms. The application and comparison of the optimization algorithms also lead to general guidelines for their applications in the development of improved multistep methods. Last but not least, theoretical analysis is also conducted on the stability of the proposed RTPEC34 method, which is corroborated in simulation experiments and thus provides general guidelines for the evaluation of realtime numerical methods. The RTPEC34 method is compared with other multistep algorithms using both numerical experiments and a real engineering example. As shown in the comparison, it achieves improved performance in terms of accuracy and stability and it is also a viable and efficient algorithm for realtime aircraft dynamics simulation.
Aircraft dynamics simulation is a complex nonlinear process whereby engine, aerodynamic, and atmospheric models are solved simultaneously [
A numerical solution of an ODE is a table of approximated values of the variables for a set of discrete time points. Multistep methods, as a class of the most widely used numerical solution, use information at more than one previous point to estimate the solution at the next point. Linear multistep methods have the form shown in (
According to the different requirements and constraints of a simulation, many researchers improved the multistep methods from different aspects. Huang [
This research, with a particular focus on the aircraft dynamics ODE problems, aims to improve the accuracy of the realtime predictorcorrector methods and analyze the stability regions of the proposed methods. Both simulation experiments of numerical examples and an engineering example are used to evaluate the performance of the proposed method by comparing its results with those obtained from the classic RK method and the methods for stiff ODEs. It has been shown in the evaluation that these proposed methods are effective. The rest of this paper is organized as follows. Section
Aircraft simulation involves a computation process that is subject to realtime constraints as it is a realtime system. This means the responses or results of the simulation process have a deadline that must be met, regardless of system load, in order for the system considered to be correct [
Evaluate
In order to improve the accuracy of the realtime predictorcorrector multistep methods, the step of the predictor equation needs to be reduced. Consider the fourthorder threestep algorithm as follows:
Additionally, based on the method of undetermined coefficients [
Thus, (
There are many optimization methods to find out the minimum value with nonlinear constraints, such as feasible direction method (FDM) [
A penalty function is generally constructed using the addition form below:
Then, the following three penalty elements are used for the GA optimization process, which are derived from (
The setting for the GA running is as follows: binary code is used; the number of individuals in the population is 1000; the crossover operator is 0.9; and mutation operator is 0.08. A simulation is done using MATLAB software, and then the parameters can be obtained for the minimum truncation error of the RTPEC34 method.
Compared with the classic optimization methods, it can be found that different parameters are obtained as shown in Table
Parameter values obtained using different optimization methods.
Parameter  FDM  QP  GA with a punishment strategy 


0.3959  0.5450  0.5181 

1.5534  0.9122  0.9963 

−0.6811  0.1233  0.0125 

0.1403  −0.0681  −0.0337 

−0.0126  0.0326  0.0249 

0.5239  0.7947  0.7425 

−0.1775  −0.3510  −0.3147 

0.0495  0.1012  0.0903 
Apply the parameters to (
The RTPEC34 method using QP optimization is shown as follows:
The RTPEC34 method using GA optimization with a punishment strategy is shown as follows:
A widely used approach to determining stability region is to apply the method to the linear ODE
Substituting
Obtaining RTPEC34 stability polynomial with the parameters shown in Table
The region of absolute stability of the RTPEC34 method.
The curves
In this section, the numerical results obtained by using the highest accuracy RTPEC34 method are presented in Example
The ODE with IVP (
The analytical solution of this ODE is
Numerical errors due to different values of the parameters in (

Formula ( 
Formula ( 
Formula (  

(FDM)  (QR)  (GA with a punishment strategy)  










0.0000 









0.0200 









0.0400 









0.0600 









0.0800 









0.1000 









0.1200 









0.1400 









0.1600 









0.1800 









0.2000 









Comparison of numerical error for (
To compare the proposed method (
Applying (
The numerical solutions obtained using the RTPEC34 method and the RK4 method.
Due to the quality of being Astable, the Rosenbrock method is a halfimplicit RK method used for solving stiff ODEs. The Rosenbrock method of order three has the following form:
The analytical solution of ODE (
Table
Comparison of numerical solutions obtained with different step sizes.

Theoretical value  Rosenbrock method  RTPEC34 method (  






0  1.0000  1.0000  1.0000  1.0000  1.0000 
0.1  1.0101  1.0075  1.0049  1.0050  1.0100 
0.2  1.0408  1.0355  1.0303  1.0405  1.0300 
0.3  1.0942  1.0859  1.0777  1.0991  1.0822 
0.4  1.1735  1.1617  1.1501  1.1843  1.1714 
0.5  1.2840  1.2679  1.2522  1.3019  1.2934 
0.6  1.4333  1.4118  1.3910  1.4601  1.4567 
0.7  1.6323  1.6038  1.5765  1.6705  1.6737 
0.8  1.8965  1.8588  1.8228  1.9498  1.9616 
0.9  2.2479  2.1978  2.1501  2.3217  2.3451 
1.0  2.7183  2.6511  2.5875  2.8203  2.8599 
Comparison of the numerical solutions obtained by the RTPEC34 and Rosenbrock methods.
The aircraft dynamics experiment is focused on the simulation of the motion model, which is a system of 12 scalar order differential equations [
where
To demonstrate the effectiveness and efficiency of the proposed method (
The nonlinear dynamics model of the F16 aircraft [
Simulation configuration for the F16 aircraft dynamics model.
Initial values of variables 



Simulation period  0~10 seconds 


Step size  0.02 seconds 
As demonstrated in these simulation experiments, the aircraft variables curves in Figure
Simulation results of different numerical methods for the F16 aircraft model.
The example of the F16 aircraft simulation demonstrates that the proposed method is adaptive in the field of dynamic ODEs. The comparison with the classic multistep methods of the same order shows the good accuracy and speed of convergence of the RTPEC34 (
In this paper, a novel realtime multistep method based on PredictEvaluateCorrect scheme of threestep fourthorder (RTPEC34) method is proposed and developed for the realtime simulation of aircraft dynamics, which is improved by obtaining the predictorcorrector parameters using optimization algorithms. The method of using GA optimization with a punishment strategy is more stable and accurate than others both theoretically and practically. This work involves theoretical analysis of the truncation error of the predictorcorrector for realtime problems as well as of the stability of the proposed RTPEC34 method. The analysis work can be used by other applications that involve improvement of accuracy and stability. Both numerical examples and engineering examples are used in simulation experiments to evaluate the performance of the RTPEC34 method by comparing it with the RK4 and the Rosenbrock methods. It is shown in the evaluation that the proposed method achieves improved performance in terms of accuracy, stability, and support of realtime simulation. Moreover, the successful application of the proposed method in the F16 aircraft experiment has shown that the proposed method is adaptive to solve the multivariable ODEs popular in aircraft dynamics simulation.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research is supported by the National Natural Science Foundation of China (no. 61374163), the National High Technology Research and Development Program (863 Program) of China (no. 2013AA041302), and the Aviation Science Foundation of China (no. 20124060179).