In this paper, a mathematical model of real-time simulation is given, and the problem of convergence on real-time Runge-Kutta algorithms is analysed. At last a theorem on the relation between the order of compensation...In this paper, a mathematical model of real-time simulation is given, and the problem of convergence on real-time Runge-Kutta algorithms is analysed. At last a theorem on the relation between the order of compensation and the convergent order of real-time algorithm is proved.展开更多
Optimization is a key technique for maximizing or minimizing functions and achieving optimal cost,gains,energy,mass,and so on.In order to solve optimization problems,metaheuristic algorithms are essential.Most of thes...Optimization is a key technique for maximizing or minimizing functions and achieving optimal cost,gains,energy,mass,and so on.In order to solve optimization problems,metaheuristic algorithms are essential.Most of these techniques are influenced by collective knowledge and natural foraging.There is no such thing as the best or worst algorithm;instead,there are more effective algorithms for certain problems.Therefore,in this paper,a new improved variant of a recently proposed metaphorless Runge-Kutta Optimization(RKO)algorithm,called Improved Runge-Kutta Optimization(IRKO)algorithm,is suggested for solving optimization problems.The IRKO is formulated using the basic RKO and local escaping operator to enhance the diversification and intensification capability of the basic RKO version.The performance of the proposed IRKO algorithm is validated on 23 standard benchmark functions and three engineering constrained optimization problems.The outcomes of IRKO are compared with seven state-of-the-art algorithms,including the basic RKO algorithm.Compared to other algorithms,the recommended IRKO algorithm is superior in discovering the optimal results for all selected optimization problems.The runtime of IRKO is less than 0.5 s for most of the 23 benchmark problems and stands first for most of the selected problems,including real-world optimization problems.展开更多
The Runge-Kutta optimiser(RUN)algorithm,renowned for its powerful optimisation capabilities,faces challenges in dealing with increasing complexity in real-world problems.Specifically,it shows deficiencies in terms of ...The Runge-Kutta optimiser(RUN)algorithm,renowned for its powerful optimisation capabilities,faces challenges in dealing with increasing complexity in real-world problems.Specifically,it shows deficiencies in terms of limited local exploration capabilities and less precise solutions.Therefore,this research aims to integrate the topological search(TS)mechanism with the gradient search rule(GSR)into the framework of RUN,introducing an enhanced algorithm called TGRUN to improve the performance of the original algorithm.The TS mechanism employs a circular topological scheme to conduct a thorough exploration of solution regions surrounding each solution,enabling a careful examination of valuable solution areas and enhancing the algorithm’s effectiveness in local exploration.To prevent the algorithm from becoming trapped in local optima,the GSR also integrates gradient descent principles to direct the algorithm in a wider investigation of the global solution space.This study conducted a serious of experiments on the IEEE CEC2017 comprehensive benchmark function to assess the enhanced effectiveness of TGRUN.Additionally,the evaluation includes real-world engineering design and feature selection problems serving as an additional test for assessing the optimisation capabilities of the algorithm.The validation outcomes indicate a significant improvement in the optimisation capabilities and solution accuracy of TGRUN.展开更多
Efficient and accurate simulation of unsteady flow presents a significant challenge that needs to be overcome in computational fluid dynamics.Temporal discretization method plays a crucial role in the simulation of un...Efficient and accurate simulation of unsteady flow presents a significant challenge that needs to be overcome in computational fluid dynamics.Temporal discretization method plays a crucial role in the simulation of unsteady flows.To enhance computational efficiency,we propose the Implicit-Explicit Two-Step Runge-Kutta(IMEX-TSRK)time-stepping discretization methods for unsteady flows,and develop a novel adaptive algorithm that correctly partitions spatial regions to apply implicit or explicit methods.The novel adaptive IMEX-TSRK schemes effectively handle the numerical stiffness of the small grid size and improve computational efficiency.Compared to implicit and explicit Runge-Kutta(RK)schemes,the IMEX-TSRK methods achieve the same order of accuracy with fewer first derivative calculations.Numerical case tests demonstrate that the IMEX-TSRK methods maintain numerical stability while enhancing computational efficiency.Specifically,in high Reynolds number flows,the computational efficiency of the IMEX-TSRK methods surpasses that of explicit RK schemes by more than one order of magnitude,and that of implicit RK schemes several times over.展开更多
The TOPKAPI （TOPographic Kinematic APproximation and Integration） model is a physically based rainfall-runoff model derived from the integration in space of the kinematic wave model. In the TOPKAPI model, rainfall-r...The TOPKAPI （TOPographic Kinematic APproximation and Integration） model is a physically based rainfall-runoff model derived from the integration in space of the kinematic wave model. In the TOPKAPI model, rainfall-runoff and runoff routing processes are described by three nonlinear reservoir differential equations that are structurally similar and describe different hydrological and hydraulic processes. Equations are integrated over grid cells that describe the geometry of the catchment, leading to a cascade of nonlinear reservoir equations. For the sake of improving the model＇s computation precision, this paper provides the general form of these equations and describes the solution by means of a numerical algorithm, the variable-step fourth-order Runge-Kutta algorithm. For the purpose of assessing the quality of the comprehensive numerical algorithm, this paper presents a case study application to the Buliu River Basin, which has an area of 3 310 km^2, using a DEM （digital elevation model） grid with a resolution of 1 km. The results show that the variable-step fourth-order Runge-Kutta algorithm for nonlinear reservoir equations is a good approximation of subsurface flow in the soil matrix, overland flow over the slopes, and surface flow in the channel network, allowing us to retain the physical properties of the original equations at scales ranging from a few meters to 1 km.展开更多
The steady state solution of long slender marine structures simply indicates the steady motion response to the excitation at top of the structure.It is very crucial especially for deep towing systems to find out how t...The steady state solution of long slender marine structures simply indicates the steady motion response to the excitation at top of the structure.It is very crucial especially for deep towing systems to find out how the towed body and towing cable work under certain towing speed.This paper has presented a direct algorithm using Runge-Kutta method for steady-state solution of long slender cylindrical structures and compared to the time iteration calculation;the direct algorithm spends much less time than the time-iteration scheme.Therefore, the direct algorithm proposed in this paper is quite efficient in providing credible reference for marine engineering applications.展开更多
In this paper, the dissipative and the forced terms of the Duffing equation are considered as the perturbations of nonlinear Hamiltonian equations and the perturbational effect is indicated by parameter ε. Firstly, b...In this paper, the dissipative and the forced terms of the Duffing equation are considered as the perturbations of nonlinear Hamiltonian equations and the perturbational effect is indicated by parameter ε. Firstly, based on the gradient- Hamiltonian decomposition theory of vector fields, by using splitting methods, this paper constructs structure-preserving algorithms （SPAs） for the Duffing equation. Then, according to the Liouville formula, it proves that the Jacobian matrix determinants of the SPAs are equal to that of the exact flow of the Duffing equation. However, considering the explicit Runge Kutta methods, this paper finds that there is an error term of order p＋l for the Jacobian matrix determinants. The volume evolution law of a given region in phase space is discussed for different algorithms, respectively. As a result, the sum of Lyapunov exponents is exactly invariable for the SPAs proposed in this paper. Finally, through numerical experiments, relative norm errors and absolute energy errors of phase trajectories of the SPAs and the Heun method （a second-order Runge-Kutta method） are compared. Computational results illustrate that the SPAs are evidently better than the Heun method when e is small or equal to zero.展开更多
A class of parallel implicit Runge-Kutta formulas is constructed for multiprocessor system. A family of parallel implicit two-stage fourth order Runge-Kutta formulas is given. For these formulas, the convergence is pr...A class of parallel implicit Runge-Kutta formulas is constructed for multiprocessor system. A family of parallel implicit two-stage fourth order Runge-Kutta formulas is given. For these formulas, the convergence is proved and the stability analysis is given. The numerical examples demonstrate that these formulas can solve an extensive class of initial value problems for the ordinary differential equations.展开更多
Based on the exact analytical solution of ordinary differential equations, a truncation of the Taylor series of the exact solution to the Nth order leads to the Nth order algebraic dynamics algorithm. A detailed numer...Based on the exact analytical solution of ordinary differential equations, a truncation of the Taylor series of the exact solution to the Nth order leads to the Nth order algebraic dynamics algorithm. A detailed numerical comparison is presented with Runge-Kutta algorithm and symplectic geometric algorithm for 12 test models. The results show that the algebraic dynamics algorithm can better preserve both geometrical and dynamical fidelity of a dynamical system at a controllable precision, and it can solve the problem of algorithm-induced dissipation for the Runge-Kutta algorithm and the problem of algorithm-induced phase shift for the symplectic geometric algorithm.展开更多
The present investigations are associated with designing Morlet wavelet neural network(MWNN)for solving a class of susceptible,infected,treatment and recovered(SITR)fractal systems of COVID-19 propagation and control....The present investigations are associated with designing Morlet wavelet neural network(MWNN)for solving a class of susceptible,infected,treatment and recovered(SITR)fractal systems of COVID-19 propagation and control.The structure of an error function is accessible using the SITR differential form and its initial conditions.The optimization is performed using the MWNN together with the global as well as local search heuristics of genetic algorithm(GA)and active-set algorithm(ASA),i.e.,MWNN-GA-ASA.The detail of each class of the SITR nonlinear COVID-19 system is also discussed.The obtained outcomes of the SITR system are compared with the Runge-Kutta results to check the perfection of the designed method.The statistical analysis is performed using different measures for 30 independent runs as well as 15 variables to authenticate the consistency of the proposed method.The plots of the absolute error,convergence analysis,histogram,performancemeasures,and boxplots are also provided to find the exactness,dependability and stability of the MWNN-GA-ASA.展开更多
In Synergetics, when a complex system evolves from one sate to another, the order parameter plays a dominant role. We can analyze the complex system state by studying the dynamic of order parameter. We developed a syn...In Synergetics, when a complex system evolves from one sate to another, the order parameter plays a dominant role. We can analyze the complex system state by studying the dynamic of order parameter. We developed a synergetic model of electricity market operation system, and studied the dynamic process of the system with empirical example, which revealed the internal mechanism of the system evolution. In order to verify the accuracy of the synergetic model, fourth-order Runge-Kutta algorithm and grey relevance method were used. Finally, we found that the reserve rate of generation was the order parameter of the system. Then we can use the principle of Synergetics to evaluate the efficiency of electricity market operation.展开更多
Malaria importation is one of the hypothetical drivers of malaria transmission dynamics across the globe.Several studies on malaria importation focused on the effect of the use of conventional malaria control strategi...Malaria importation is one of the hypothetical drivers of malaria transmission dynamics across the globe.Several studies on malaria importation focused on the effect of the use of conventional malaria control strategies as approved by the World Health Organization(WHO)on malaria transmission dynamics but did not capture the effect of the use of traditional malaria control strategies by vigilant humans.In order to handle the aforementioned situation,a novel system of Ordinary Differential Equations(ODEs)was developed comprising the human and the malaria vector compartments.Analysis of the system was carried out to assess its quantitative properties.The novel computational algorithm used to solve the developed system of ODEs was implemented and benchmarked with the existing Runge-Kutta numerical solution method.Furthermore,simulations of different vigilant conditions useful to control malaria were carried out.The novel system of malaria models was well-posed and epidemiologically meaningful based on its quantitative properties.The novel algorithm performed relatively better in terms of model simulation accuracy than Runge-Kutta.At the best model-fit condition of 98%vigilance to the use of conventional and traditional malaria control strategies,this study revealed that malaria importation has a persistent impact on malaria transmission dynamics.In lieu of this,this study opined that total vigilance to the use of the WHO-approved and traditional malaria management tools would be the most effective control strategy against malaria importation.展开更多
文摘In this paper, a mathematical model of real-time simulation is given, and the problem of convergence on real-time Runge-Kutta algorithms is analysed. At last a theorem on the relation between the order of compensation and the convergent order of real-time algorithm is proved.
基金The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University,Saudi Arabia,for funding this work through the Research Group Program under Grant No:RGP.2/108/42.
文摘Optimization is a key technique for maximizing or minimizing functions and achieving optimal cost,gains,energy,mass,and so on.In order to solve optimization problems,metaheuristic algorithms are essential.Most of these techniques are influenced by collective knowledge and natural foraging.There is no such thing as the best or worst algorithm;instead,there are more effective algorithms for certain problems.Therefore,in this paper,a new improved variant of a recently proposed metaphorless Runge-Kutta Optimization(RKO)algorithm,called Improved Runge-Kutta Optimization(IRKO)algorithm,is suggested for solving optimization problems.The IRKO is formulated using the basic RKO and local escaping operator to enhance the diversification and intensification capability of the basic RKO version.The performance of the proposed IRKO algorithm is validated on 23 standard benchmark functions and three engineering constrained optimization problems.The outcomes of IRKO are compared with seven state-of-the-art algorithms,including the basic RKO algorithm.Compared to other algorithms,the recommended IRKO algorithm is superior in discovering the optimal results for all selected optimization problems.The runtime of IRKO is less than 0.5 s for most of the 23 benchmark problems and stands first for most of the selected problems,including real-world optimization problems.
基金Natural Science Foundation of Zhejiang Province,Grant/Award Numbers:LTGS23E070001,LZ22F020005,LTGY24C060004National Natural Science Foundation of China,Grant/Award Numbers:62076185,62301367,62273263。
文摘The Runge-Kutta optimiser(RUN)algorithm,renowned for its powerful optimisation capabilities,faces challenges in dealing with increasing complexity in real-world problems.Specifically,it shows deficiencies in terms of limited local exploration capabilities and less precise solutions.Therefore,this research aims to integrate the topological search(TS)mechanism with the gradient search rule(GSR)into the framework of RUN,introducing an enhanced algorithm called TGRUN to improve the performance of the original algorithm.The TS mechanism employs a circular topological scheme to conduct a thorough exploration of solution regions surrounding each solution,enabling a careful examination of valuable solution areas and enhancing the algorithm’s effectiveness in local exploration.To prevent the algorithm from becoming trapped in local optima,the GSR also integrates gradient descent principles to direct the algorithm in a wider investigation of the global solution space.This study conducted a serious of experiments on the IEEE CEC2017 comprehensive benchmark function to assess the enhanced effectiveness of TGRUN.Additionally,the evaluation includes real-world engineering design and feature selection problems serving as an additional test for assessing the optimisation capabilities of the algorithm.The validation outcomes indicate a significant improvement in the optimisation capabilities and solution accuracy of TGRUN.
基金supported by the National Natural Science Foundation of China(No.92252201)the Fundamental Research Funds for the Central Universitiesthe Academic Excellence Foundation of Beihang University(BUAA)for PhD Students。
文摘Efficient and accurate simulation of unsteady flow presents a significant challenge that needs to be overcome in computational fluid dynamics.Temporal discretization method plays a crucial role in the simulation of unsteady flows.To enhance computational efficiency,we propose the Implicit-Explicit Two-Step Runge-Kutta(IMEX-TSRK)time-stepping discretization methods for unsteady flows,and develop a novel adaptive algorithm that correctly partitions spatial regions to apply implicit or explicit methods.The novel adaptive IMEX-TSRK schemes effectively handle the numerical stiffness of the small grid size and improve computational efficiency.Compared to implicit and explicit Runge-Kutta(RK)schemes,the IMEX-TSRK methods achieve the same order of accuracy with fewer first derivative calculations.Numerical case tests demonstrate that the IMEX-TSRK methods maintain numerical stability while enhancing computational efficiency.Specifically,in high Reynolds number flows,the computational efficiency of the IMEX-TSRK methods surpasses that of explicit RK schemes by more than one order of magnitude,and that of implicit RK schemes several times over.
基金supported by the National Natural Science Foundation of China(Grant No.50479017)the Program for Changjiang Scholars and Innovative Research Teams in Universities(Grant No.IRT071)
文摘The TOPKAPI （TOPographic Kinematic APproximation and Integration） model is a physically based rainfall-runoff model derived from the integration in space of the kinematic wave model. In the TOPKAPI model, rainfall-runoff and runoff routing processes are described by three nonlinear reservoir differential equations that are structurally similar and describe different hydrological and hydraulic processes. Equations are integrated over grid cells that describe the geometry of the catchment, leading to a cascade of nonlinear reservoir equations. For the sake of improving the model＇s computation precision, this paper provides the general form of these equations and describes the solution by means of a numerical algorithm, the variable-step fourth-order Runge-Kutta algorithm. For the purpose of assessing the quality of the comprehensive numerical algorithm, this paper presents a case study application to the Buliu River Basin, which has an area of 3 310 km^2, using a DEM （digital elevation model） grid with a resolution of 1 km. The results show that the variable-step fourth-order Runge-Kutta algorithm for nonlinear reservoir equations is a good approximation of subsurface flow in the soil matrix, overland flow over the slopes, and surface flow in the channel network, allowing us to retain the physical properties of the original equations at scales ranging from a few meters to 1 km.
基金the National Natural Science Foundation of China(Nos.51009092 and 50909061)the Doctoral Foundation of Education Ministry of China (No.20090073120013)the National High Technology Research and Development Program (863) of China (No.2008AA092301-1)
文摘The steady state solution of long slender marine structures simply indicates the steady motion response to the excitation at top of the structure.It is very crucial especially for deep towing systems to find out how the towed body and towing cable work under certain towing speed.This paper has presented a direct algorithm using Runge-Kutta method for steady-state solution of long slender cylindrical structures and compared to the time iteration calculation;the direct algorithm spends much less time than the time-iteration scheme.Therefore, the direct algorithm proposed in this paper is quite efficient in providing credible reference for marine engineering applications.
基金Project supported by the National Natural Science Foundation of China (Grant No 10572021)the Doctoral Programme Foundation of Institute of Higher Education of China (Grant No 20040007022)
文摘In this paper, the dissipative and the forced terms of the Duffing equation are considered as the perturbations of nonlinear Hamiltonian equations and the perturbational effect is indicated by parameter ε. Firstly, based on the gradient- Hamiltonian decomposition theory of vector fields, by using splitting methods, this paper constructs structure-preserving algorithms （SPAs） for the Duffing equation. Then, according to the Liouville formula, it proves that the Jacobian matrix determinants of the SPAs are equal to that of the exact flow of the Duffing equation. However, considering the explicit Runge Kutta methods, this paper finds that there is an error term of order p＋l for the Jacobian matrix determinants. The volume evolution law of a given region in phase space is discussed for different algorithms, respectively. As a result, the sum of Lyapunov exponents is exactly invariable for the SPAs proposed in this paper. Finally, through numerical experiments, relative norm errors and absolute energy errors of phase trajectories of the SPAs and the Heun method （a second-order Runge-Kutta method） are compared. Computational results illustrate that the SPAs are evidently better than the Heun method when e is small or equal to zero.
基金Project supported by the National Natural Science Foundation of China
文摘A class of parallel implicit Runge-Kutta formulas is constructed for multiprocessor system. A family of parallel implicit two-stage fourth order Runge-Kutta formulas is given. For these formulas, the convergence is proved and the stability analysis is given. The numerical examples demonstrate that these formulas can solve an extensive class of initial value problems for the ordinary differential equations.
基金Supported by the National Natural Science Foundation of China (Grant Nos. 10375039 and 90503008)the Doctoral Program Foundation from the Ministry of Education of China,and the Center of Nuclear Physics of HIRFL of China
文摘Based on the exact analytical solution of ordinary differential equations, a truncation of the Taylor series of the exact solution to the Nth order leads to the Nth order algebraic dynamics algorithm. A detailed numerical comparison is presented with Runge-Kutta algorithm and symplectic geometric algorithm for 12 test models. The results show that the algebraic dynamics algorithm can better preserve both geometrical and dynamical fidelity of a dynamical system at a controllable precision, and it can solve the problem of algorithm-induced dissipation for the Runge-Kutta algorithm and the problem of algorithm-induced phase shift for the symplectic geometric algorithm.
基金The authors extend their appreciation to the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University for funding this work through Research Group No.RG-21-09-12.
文摘The present investigations are associated with designing Morlet wavelet neural network(MWNN)for solving a class of susceptible,infected,treatment and recovered(SITR)fractal systems of COVID-19 propagation and control.The structure of an error function is accessible using the SITR differential form and its initial conditions.The optimization is performed using the MWNN together with the global as well as local search heuristics of genetic algorithm(GA)and active-set algorithm(ASA),i.e.,MWNN-GA-ASA.The detail of each class of the SITR nonlinear COVID-19 system is also discussed.The obtained outcomes of the SITR system are compared with the Runge-Kutta results to check the perfection of the designed method.The statistical analysis is performed using different measures for 30 independent runs as well as 15 variables to authenticate the consistency of the proposed method.The plots of the absolute error,convergence analysis,histogram,performancemeasures,and boxplots are also provided to find the exactness,dependability and stability of the MWNN-GA-ASA.
文摘In Synergetics, when a complex system evolves from one sate to another, the order parameter plays a dominant role. We can analyze the complex system state by studying the dynamic of order parameter. We developed a synergetic model of electricity market operation system, and studied the dynamic process of the system with empirical example, which revealed the internal mechanism of the system evolution. In order to verify the accuracy of the synergetic model, fourth-order Runge-Kutta algorithm and grey relevance method were used. Finally, we found that the reserve rate of generation was the order parameter of the system. Then we can use the principle of Synergetics to evaluate the efficiency of electricity market operation.
基金funded by the National Institutes of Health Common Fund [Grant no:1U2RTW010679]under the West African Sustainable Leadership and Innovation Training in Bioinformatics Research Project and the World Bank (2019e2024).
文摘Malaria importation is one of the hypothetical drivers of malaria transmission dynamics across the globe.Several studies on malaria importation focused on the effect of the use of conventional malaria control strategies as approved by the World Health Organization(WHO)on malaria transmission dynamics but did not capture the effect of the use of traditional malaria control strategies by vigilant humans.In order to handle the aforementioned situation,a novel system of Ordinary Differential Equations(ODEs)was developed comprising the human and the malaria vector compartments.Analysis of the system was carried out to assess its quantitative properties.The novel computational algorithm used to solve the developed system of ODEs was implemented and benchmarked with the existing Runge-Kutta numerical solution method.Furthermore,simulations of different vigilant conditions useful to control malaria were carried out.The novel system of malaria models was well-posed and epidemiologically meaningful based on its quantitative properties.The novel algorithm performed relatively better in terms of model simulation accuracy than Runge-Kutta.At the best model-fit condition of 98%vigilance to the use of conventional and traditional malaria control strategies,this study revealed that malaria importation has a persistent impact on malaria transmission dynamics.In lieu of this,this study opined that total vigilance to the use of the WHO-approved and traditional malaria management tools would be the most effective control strategy against malaria importation.
