A Mathematical Model of Real-Time Simulation and the Convergence Analysis on Real-Time Runge-Kutta Algorithms

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.

IRKO:An Improved Runge-Kutta Optimization Algorithm for Global Optimization Problems

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.

Numerical algorithm of distributed TOPKAPI model and its application

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.

Structure-preserving algorithms for the Duffng equation

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.

A Class of Parallel Implicit Runge-Kutta Formulas

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.

Algebraic dynamics algorithm:Numerical comparison with Runge-Kutta algorithm and symplectic geometric algorithm

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.

Numerical Computational Heuristic Through Morlet Wavelet Neural Network for Solving the Dynamics of Nonlinear SITR COVID-19

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.

Study on the Synergetic Mechanism for the Dynamic Evaluation of Electricity Market Operational Efficiency

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.

Direct Algorithms for Steady-State Solution of Long Slender Marine Structures

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.

A Novel Class of Energy-Preserving Runge-Kutta Methods for the Korteweg-de Vries Equation

In this paper,we present a quadratic auxiliary variable approach to develop a new class of energy-preserving Runge-Kutta methods for the Korteweg-de Vries equation.The quadratic auxiliary variable approach is first proposed to reformulate the original model into an equivalent system,which transforms the energy conservation law of the Korteweg-de Vries equation into two quadratic invariants of the reformulated system.Then the symplectic Runge-Kutta methods are directly employed for the reformulated model to arrive at a new kind of time semi-discrete schemes for the original problem.Under consistent initial conditions,the proposed methods are rigorously proved to maintain the original energy conservation law of the Korteweg-de Vries equation.In addition,the Fourier pseudo-spectral method is used for spatial discretization,resulting in fully discrete energy-preserving schemes.To implement the proposed methods effectively,we present a very efficient iterative technique,which not only greatly saves the calculation cost,but also achieves the purpose of practically preserving structure.Ample numerical results are addressed to confirm the expected order of accuracy,conservative property and efficiency of the proposed algorithms.

Mathematical modeling of malaria transmission dynamics in humans with mobility and control states

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.