Although predictor-corrector methods have been extensively applied, they might not meet the requirements of practical applications and engineering tasks, particularly when high accuracy and efficiency are necessary. A novel class of correctors based on feedback-accelerated Picard iteration (FAPI) is proposed to further enhance computational performance. With optimal feedback terms that do not require inversion of matrices, significantly faster convergence speed and higher numerical accuracy are achieved by these correctors compared with their counterparts; however, the computational complexities are comparably low. These advantages enable nonlinear engineering problems to be solved quickly and accurately, even with rough initial guesses from elementary predictors. The proposed method offers flexibility, enabling the use of the generated correctors for either bulk processing of collocation nodes in a domain or successive corrections of a single node in a finite difference approach. In our method, the functional formulas of FAPI are discretized into numerical forms using the collocation approach. These collocated iteration formulas can directly solve nonlinear problems, but they may require significant computational resources because of the manipulation of high-dimensional matrices. To address this, the collocated iteration formulas are further converted into finite difference forms, enabling the design of lightweight predictor-corrector algorithms for real-time computation. The generality of the proposed method is illustrated by deriving new correctors for three commonly employed finite-difference approaches: the modified Euler approach, the Adams-Bashforth-Moulton approach, and the implicit Runge-Kutta approach. Subsequently, the updated approaches are tested in solving strongly nonlinear problems, including the Matthieu equation, the Duffing equation, and the low-earth-orbit tracking problem. The numerical findings confirm the computational accuracy and efficiency of the derived predictor-corrector algorithms.

In real-world engineering tasks and scientific research, predictor-corrector methods are fundamental to numerically solving nonlinear differential equations [

This study presents three main contributions. First, an approach to derive fast and accurate correctors to enhance the performance of predictor-corrector methods for more advanced tasks in real life is proposed. This is achieved by replacing PI with feedback-accelerated Picard iteration (FAPI) [

The applications of the enhanced predictor-corrector methods and FAPI span various domains, including simulations of n-body problems in celestial mechanics [

This study is structured as follows. In

Consider a nonlinear dynamical system in the form of

Without loss of generality, we suppose

It can be rewritten as

where

Suppose

Let

Now we want to make all the components of the function

Then we collect the terms including

Note that the boundary value of

from which we can derive the first order appoximation of

where

which means the error caused will not exceed

By substituting

where

Previously, the formula of FAPI takes the form of

with

Simply approximating

By integrating both sides of

For clarity, the two versions of FAPI fomulae are listed in

Versions | Formula |
---|---|

1st | |

2nd |

By collocating

The block diagonal matrices

We suppose that

Using the approximation in

where

Further, by making the basis functions in

where the integration matrix

For simplicity, we denote

However, it may cause deterioration of the numerical solution, because

where the modified integration matrix

Denoting

We further denote the constant matrix

where

Further numerical discretization is made to the collocation form of

In

We suppose that

Similar approximations can be made to the components

Using the approximations in

and

where the integration matrix

For simplicity, we denote

where

For clarity, the two versions of collocated FAPI formulae are displayed in

Versions | Formula |
---|---|

1st | |

2nd |

Each row of

where the superscript

Given the values of collocation nodes

To apply FAPI as numerical corrector in finite difference method, we need to use the same collocation nodes and basis functions underlying the finite difference discretization. The relationship between finite difference and collocation is briefly reviewed herein.

The collocation form of

where

Besides,

The collocation form of

where

It is an implicit finite difference formula. Usually, we let

Since

Herein, we consider the Modified Euler method (MEM) that only uses 2 nodes, of which one is known and the other is unknown. The collocation counterpart of this method has 2 collocation nodes, and takes the Lagrange polynomials

and

By substituting

The first row of

where

However, the modified Euler method utilizes the trapezoidal formula instead of the implicit Euler formula to make correction. The trapezoidal formula can be derived from

and

Note that we start the integration at

The second row of

Now we replace

With

Thus

According to

It is rewritten as

where

According to

It is rewritten as

where

Taking the 4-th order Adams-Bashforth-Moulton (ABM) method for instance, it solves

of which

To replace the Picard iteration with Feedback-Accelerated Picard iteration in the Adams-Moulton corrector (

With

To keep consistent with the Adams-Moulton formula underlying

and

They are further expressed as

and

Then the matrix

The matrix

By substituting

It is explicitly expressed as

where the terms related with

According to

It is rewritten as

The equivalence between implicit Runge-Kutta (IRK) method and collocation method has been well discussed and proved [

By re-scaling the time segment as

where

From

The 4-stage implicit Runge-Kutta method using Picard iteration is expressed as

By substituting

where

The 2nd version of the 4-stage IRK-FAPI formula can be obtained by substituting

In this section, conventional predictor-corrector methods, the corresponding FAPI enhanced methods, and MATLAB-built-in ode45/ode113 were used to solve the dynamical responses of some typical nonlinear systems, such as the Mathieu equation, the forced Duffing equation, and the perturbed two-body problem. The numerical simulation was conducted in MATLAB R2017a using an ASUS laptop with an Intel Core i5-7300HQ CPU. GPU acceleration and parallel processing were not used in the following examples.

In this study, to conduct a comprehensive and rational evaluation of each approach, two approaches were employed to compare the accuracy, convergence speed, and computational efficiency of conventional approaches and the corresponding FAPI-enhanced methods. The first approach involves performing the iterative correction only once, which is the general usage of the prediction-correction algorithm. The second approach involves repeating the iterative correction until the corresponding terminal conditions are met.

The maximum computational error in the simulation time interval was selected to represent the computational accuracy when evaluating the computational accuracy of each approach. The solution of ode45 or ode113 was employed as the benchmark for calculating this computational error. The relative and absolute errors of ode45 and ode113 were set as

The Mathieu equation is

where the parameters are set as

The forced Duffing equation is

where the parameters are set as

The orbital problem is considered in the following form:

where

The computational accuracy of orbital motion plays a crucial role in determining the success of space missions. In the case of LEO, the long-term trajectory of a spacecraft can be significantly affected by a very small perturbation term of gravity force. A high-order gravitational model should be used to achieve relatively high accuracy. In this case, most computational time is dedicated to assessing the gravitational force terms. The relative error of position is defined as

Using ode45/ode113 as the benchmark,

Problems | Corrected once | Corrected until converged |
---|---|---|

MEM/ME-FAPI | MEM/ME-FAPI | |

Mathieu equation | ||

The forced duffing equation | ||

Low-earth-orbit tracking problem |

The MEM and ME-FAPI methods exhibited a good match with ode45/ode113. In

A more comprehensive presentation of the computational accuracy and efficiency of each method is shown in

The following section examines the performance indices of each approach after the convergence of the iterative process, including the maximum computational error, average iteration steps, and computational efficiency.

Furthermore,

When corrected only once, the simulation results in

Problems | Corrected once | Corrected until converged |
---|---|---|

ABM/ABM-FAPI | ABM/ABM-FAPI | |

Mathieu equation | ||

The forced duffing equation | ||

Low-earth-orbit tracking problem | / |

New results can be obtained when the algorithms converge, as shown in

The simulation findings indicate that, in all cases, the 1st ABM-FAPI method significantly reduces computational time compared with the classical ABM method. For instance, as shown in

Furthermore, the algorithm complexity of the 2nd ABM-FAPI method, which incorporates more information about the Jacobi matrix at time nodes, is significantly higher than that of the 1st ABM-FAPI method. However, as shown in

The initial approximation in each step is selected as a straight line without additional computation of the derivatives when using the IRK method and the corresponding IRK-FAPI method to solve the nonlinear system. This initial approximation serves as a “cold start” for the methods. Generally, the “cold start” is a very rough prediction method.

Problems | Corrected once | Corrected until converged |
---|---|---|

IRK/IRK-FAPI | IRK/IRK-FAPI | |

Mathieu equation | ||

The forced duffing equation | ||

Low-earth-orbit tracking problem | / |

Similarly,

This study proposes a novel approach for deriving efficient predictor-corrector approaches based on FAPI. Three classical approaches, including the modified Euler approach, the Adams-Bashforth-Moulton approach, and the implicit Runge-Kutta approach, are used to demonstrate how FAPI can be employed to modify and improve commonly used predictor-corrector approaches. For each classical method, two versions of the FAPI correctors are derived. The resulting six FAPI-featured approaches are further used to numerically solve three different types of nonlinear dynamical systems originating from various research fields, ranging from mechanical vibration to astrodynamics. The numerical findings indicate that these enhanced correctors can effectively solve these problems with high accuracy and efficiency, with the 1st version of FAPI featured methods displaying superior performance in most cases. For instance, the 1st FAPI correctors achieve 1–2 orders of magnitude higher accuracy and efficiency than conventional correctors in solving the Mathieu equation. Furthermore, at least one version of the FAPI correctors shows superiority over the original correctors in all these problems, particularly when the correctors are used only once per step. In future studies, the application of the proposed approach to bifurcation problems and high-dimensional nonlinear structural problems will be presented.

The proposed approach offers a general framework for enhancing existing predictor-corrector methods by deriving new correctors using FAPI. Although three existing methods were retrofitted and their improvements demonstrated with three problems, it is anticipated that similar enhancements can be achieved by applying the proposed approach to various other predictor-corrector methods and nonlinear problems. For clarity, methods with low approximation orders (<5) were selected to illustrate our approach. However, methods of higher order can also be derived using the same procedure. In addition, our method can be directly combined with existing adaptive computational approaches for predictor-corrector methods to adaptively solve real-world problems.

The authors are grateful for the support by Northwestern Polytechnical University and Texas Tech University. Additionally, we would like to express our appreciation to anonymous reviewers and journal editors for assistance.

This work is supported by the Fundamental Research Funds for the Central Universities (No. 3102019HTQD014) of Northwestern Polytechnical University, Funding of National Key Laboratory of Astronautical Flight Dynamics, and Young Talent Support Project of Shaanxi State.

The authors confirm contribution to the paper as follows: study conception and design: Xuechuan Wang, Wei He; data collection: Wei He; analysis and interpretation of results: Xuechuan Wang, Wei He, Haoyang Feng; draft manuscript preparation: Xuechuan Wang, Wei He, Haoyang Feng, Satya N. Atluri. All authors reviewed the results and approved the final version of the manuscript.

The data supporting the conclusions of this article are included within the article. Any queries regarding these data may be directed to the corresponding author.

The authors declare that they have no conflicts of interest to report regarding the present study.