Modified Runge Kutta Research Articles

Stability and Optimality Criteria for an SVIR Epidemic Model with Numerical Simulation

The mathematical modeling of infectious diseases plays a vital role in understanding and predicting disease transmission, as underscored by recent global outbreaks; to delve deep into the dynamic of infectious disease considering latent period presciently is inevitable as it bridges the gap between realistic nature and mathematical modeling. This study extended the classical Susceptible–Infected–Recovered (SIR) model by incorporating vaccination strategies during incubation. We introduced multiple time delays to an account incubation period to capture realistic disease dynamics better. The model is formulated as a system of delay differential equations that describe the transmission dynamics of diseases such as polio or COVID-19, or diseases for which vaccination exists. Critical aspects of the study include proving the positivity of the model’s solutions, calculating the basic reproduction number (R0) using next-generation matrix theory, and identifying disease-free and endemic equilibrium points. The local stability of these equilibria is then analyzed using the Routh–Hurwitz criterion. Due to the complexity introduced by the delay components, we examine the stability by studying the roots of a fourth-degree exponential polynomial. The effects of educational campaigns and vaccination efficacy are also investigated as control measures. Furthermore, an optimization problem is formulated, based on Pontryagin’s maximum principle, to minimize the number of infections and associated intervention costs. Numerical simulations of the delay differential equations are conducted, and a modified Runge–Kutta method with delays is used to solve the optimal control problem. Finally, we present a few simulation results to illustrate the analytical findings.

A modified Runge–Kutta optimization for optimal photovoltaic and battery storage allocation under uncertainty and load variation

The interest in incorporating environmentally friendly and renewable sources of energy, like photovoltaic (PV) technology, into electricity grids has grown significantly. These sources offer benefits, such as reduced power losses and improved voltage stability. To optimize these advantages, it is essential to determine optimal placement and management of these energy resources. This paper proposes an Improved RUNge–Kutta optimizer (IRUN) for allocating PV-based distributed generations (DGs) and Battery Energy Storage (BES) in distribution networks. IRUN utilizes three strategies to avoid local optima and enhance exploration and exploitation phases: a non-linear operator for smoother transitions, a Chaotic Local Search for thorough exploration, and diverse solution updates for refinement. The efficacy of IRUN is evaluated using 10 benchmark functions from the CEC’20 test suite, followed by statistical analysis. Next, IRUN is used to optimize the allocation of PVDG and BES to minimize energy losses in two standard IEEE distribution networks. The optimization problem is divided into two stages. In the first stage, the optimal size and the location of PV systems are calculated to meet peak load demand. In the second stage, considering time-varying load demand and intermittent PV generation, effective energy management of BES is employed. The effectiveness of IRUN is compared against the original RUN and other well-known optimization algorithms through simulation results. The comprehensive analysis demonstrates that IRUN outperforms the compared algorithms, making it a leading solution for optimizing PV distributed generation and BES allocation in distribution networks and the results show that the energy loss reduction reaches 63.54% and 68.19% when using PVand BES in IEEE 33-bus and IEEE 69 bus respectively.

A modified Runge–Kutta method for increasing stability properties

In control theory, we are interested mainly in investigating asymptotic stability of systems. In this paper explicit Runge–Kutta methods are investigated for numerical solutions of nonlinear ODEs with known strict Lyapunov functions. In order to increase stability properties, a modified version of explicit Runge–Kutta methods is presented. The modified Runge–Kutta method is explicit. It is different from the projection methods in literature which are implicit. Comparing with the standard Runge–Kutta method, the modified Runge–Kutta one has better stability properties. Numerical examples are provided to illustrate the effectiveness of the modified Runge–Kutta method.

A Modified Runge–Kutta Scheme for the Generalized Benjamin–Bona–Mahony–Burgers Equation

This paper presents a modified Runge–Kutta (MRK) scheme for the generalized Benjamin–Bona–Mahony–Burgers equation. The MRK method separate this equation into two parts which apply different finite difference scheme in space, and finally use fourth-order Runge–Kutta method in time.The scheme is conditionally stable and convergent with order of O (тау4+h2), the Comparisons of the existing numerical methods demonstrate that this scheme is capable of high accuracy and stability.

A Modified Runge–Kutta Scheme for the Generalized Benjamin–Bona–Mahony–Burgers Equation

A Modified Runge–Kutta Scheme for the Generalized Benjamin–Bona–Mahony–Burgers Equation

Stochastic optimal power flow analysis of power systems with wind/PV/ TCSC using a developed Runge Kutta optimizer

Recently, renewable energy resources such as wind turbines (WTs) and photovoltaic (PV) systems are wildly installed in electrical systems. However, the main challenge that related to these resources is the optimal planning and operation of the power system due to their stochastic natural and intermittency. However, OPF solution in case of considering the uncertainties of RERs and optimal integration of FACTS device became more complex and not easy task. Consequently, this paper proposes a modified version of the RUNge Kutta optimizer (MRUN) for solving the stochastic OPF problem with optimal integration of the WTs and PV systems along with Thyristor controlled series compensator (TCSC). The Modified RUNge Kutta optimizer (MRUN) is based on Cauchy mutation and the quasi-oppositional based learning techniques to render the RUN optimization technique to be more robust and powerful. The objective functions of the current optimization problem are the total expected power losses (TEPL), the total expected voltage deviations (TEVD) and the total expected voltage stability (TEVS). The proposed algorithm is tested on IEEE 57-bus system. After the RERs and TCSCs integration under uncertainty state, the reduction in the TEPL and TEVD are 24.75 %, and 14.74 % while the TEVS is enhanced by 18.33 % compared to based case (without RERs and TCSCs). In addition to that the proposed MRUN is superior for solving the OPF and assigning the optimal sites and sizes of the RERs and the TCSCs compared to the conventional RUN.

Error analysis of modified Runge–Kutta–Nyström methods for nonlinear second-order delay boundary value problems

This paper is concerned with the numerical solutions of nonlinear second-order boundary value problems with time-variable delay. By adapting Runge–Kutta–Nyström (RKN) methods and combining Lagrange interpolation, a class of modified RKN (MRKN) methods are suggested for solving the problems. Under some suitable conditions, MRKN methods are proved to be convergent of order min{p,q}, where p,q are the local orders of MRKN methods and Lagrange interpolation, respectively. Numerical experiments further confirm the computational effectiveness and accuracy of MRKN methods.

Optimal PID plus second-order derivative controller design for AVR system using a modified Runge Kutta optimizer and Bode’s ideal reference model

Optimal PID plus second-order derivative controller design for AVR system using a modified Runge Kutta optimizer and Bode’s ideal reference model

Optimal reconfiguration strategy based on modified Runge Kutta optimizer to mitigate partial shading condition in photovoltaic systems

In this article, a new variant of a recent optimization algorithm namely Runge Kutta optimizer (RUN) is proposed to solve the partial shading condition in photovoltaic systems and global optimization. The RUN’s improvement is mainly to mitigate the lack of the solutions quality, the imbalance between the exploitation and exploration phases, and premature convergence of the RUN algorithm. The mRUN, in which the standard RUN is incorporated with a promising strategy namely Orthogonal learning (OL) strategy to address out the original RUN’s drawbacks. To estimate the proposed algorithm’s efficiency, two experimental series were performed. In the first experiment, the mRUN was compared with other state-of-art metaheuristics on IEEE CEC’2020 test suite. Moreover, the quantitative and qualitative approved the robustness of the suggested algorithm. As a second experimental series, the proposed approach is investigated on an application of renewable energy-based system which is reconfiguration of partially shaded photovoltaic (PV) array. The main target of this process is to enhance the generated power from the PV array. Two shade patterns are investigated on 9 × 9 array and the obtained results via the modified RUN are compared to total cross tied (TCT) and SudoKu and other metaheuristic-based arrangements of Aquila optimizer (AO), Harris hawks optimizer (HHO), and Runge Kutta optimizer (RUN). The generated power from the PV array, arranged via the proposed approach, is enhanced by 28.41% and 1.015% in the first shade pattern compared to the TCT and conventional RUN configurations, respectively. In the second shade pattern, the proposed mRUN succeeded in improving the extracted power by 40.32% and 0.29% compared to TCT and RUN configurations, respectively. The modified RUN performed well in both studied cases outperforming the others in achieving the most enhanced power from the array.

High-Performance Computing Probabilistic Fracture Mechanics Implementation for Gas Turbine Rotor Disks on Distributed Architectures Including Graphics Processing Units

Abstract We present an efficient Monte Carlo-based probabilistic fracture mechanics simulation implementation for heterogeneous high-performance computing (HPC) architectures including central processing units (CPUs) and graphics processing units (GPUs). The specific application focuses on large heavy-duty gas turbine rotor components for the energy sector. A reliable probabilistic risk quantification requires the simulation of millions to billions of Monte Carlo (MC) samples. We apply a modified Runge–Kutta algorithm in order to solve numerically the fatigue crack growth for this large number of cracks for varying initial crack sizes, locations, material, and service conditions. This compute intensive simulation has already been demonstrated to perform efficiently and scalable on parallel and distributed HPC architectures including hundreds of CPUs utilizing the message passing interface (MPI) paradigm. In this work, we go a step further and include GPUs in our parallelization strategy. We develop a load distribution scheme to share one or more GPUs on compute nodes distributed over a network. We detail technical challenges and solution strategies in performing the simulations on GPUs efficiently. We show that the key computation of the modified Runge–Kutta integration step speeds up over two orders of magnitude on a typical GPU compared to a single threaded CPU. This is supported by our use of GPU textures for efficient interpolation of multidimensional tables utilized in the implementation. We demonstrate weak and strong scaling of our GPU implementation, i.e., that we can efficiently utilize a large number of GPUs/CPUs in order to solve for more MC samples, or reduce the computational turn-around time, respectively. On seven different GPUs spanning four generations, the presented probabilistic fracture mechanics simulation tool ProbFM achieves a speed-up ranging from 16.4× to 47.4× compared to single threaded CPU implementation.

Optimum Integration Procedure for Connectionist and Dynamic Field Equations.

Connectionist and dynamic field models consist of a set of coupled first-order differential equations describing the evolution in time of different units. We compare three numerical methods for the integration of these equations: the Euler method, and two methods we have developed and present here: a modified version of the fourth-order Runge Kutta method, and one semi-analytical method. We apply them to solve a well-known nonlinear connectionist model of retrieval in single-digit multiplication, and show that, in many regimes, the semi-analytical and modified Runge Kutta methods outperform the Euler method, in some regimes by more than three orders of magnitude. Given the outstanding difference in execution time of the methods, and that the EM is widely used, we conclude that the researchers in the field can greatly benefit from our analysis and developed methods.

Solution of Maxwell’s Equations for Nonrectangular Electromagnetic Applications

The use of a fourth-order modified Runge–Kutta (MRK) scheme on a transformed coordinate system with Maxwell’s equations for nonrectangular domains and applications is presented. Maxwell’s equations are the governing equations for modeling electromagnetic wave propagation involving scattering, radiating structures, transmission lines, radar, biomedical applications, and nondestructive testing. Because of complex geometries in most problems where the material boundaries are not parallel to the grid axis, the application of finite differencing schemes in physical coordinates becomes nonviable. Therefore, by transforming the arbitrary-shaped structures to a uniform rectangular grid, numerical schemes and boundary conditions can be easily implemented. Numerical results for four cases are presented in this paper. Accuracy of the scheme has been established by comparing numerical results with the exact solution and error distribution plots. In the third case, scattering from the lossless and the lossy square cylindrical dielectric device where the plane wave source is injected using the total-field–scattered-field technique has been investigated. The results are compared with the results obtained from the benchmark finite difference time domain scheme. Lastly, the application of the numerical model to simulate scattering from curved boundaries has been presented in the fourth example.

Comparative Study on Various Numerical Schemes for Solving Nozzle Flow Problems

The inviscid compressible flows represent the flow model of high-speed flow with the viscous effects can be ignored. Such flow phenomena appear when a high Reynolds number flow past trough a streamlined body at relative a low angle attack. The most commercial passenger airplane is operated at such flow condition, as results the aircraft industries have already put their efforts to develop the capability for solving the governing equation of inviscid compressible flow. The governing equation of inviscid compressible flow which is called as the Euler Equation can solved numerically. There are various numerical schemes had been developed and the most of them are dedicated for capturing the presence of shock wave phenomena which is normally appearing at high speed flow. The present work presents a comparative study over various numerical schemes which they are already been developed for solving the Euler equations. Those numerical schemes are the MacCormack Scheme, the Fourth Order Runge Kutta Scheme, The First and Second Order Flux Splitting Steger Warming Scheme, Flux Splitting Bram Van Leer Scheme, the Harten Yee TVD scheme, Roe Sweby TVD scheme, Davies – Yee TVD scheme and the Modified Four Order Runge Kutta and Harten Yee TVD scheme. These nine numerical schemes are applied to the case of flow past through nozzle. The flow phenomena may appear along the nozzle may be the form completely as an isentropic flow or the flow with shock wave. These two flow phenomena are depended the boundary condition of the flow problem under considered. The implementation to case of isentropic flow along the nozzle, those nine numerical schemes works well, and the number of iterations required nearly the same between one with the other except for the Fourth Order Runge Kutta Scheme. In the case of flow with shock wave, the Fourth Order Runge Kutta Scheme, Second Order Steger Warming and the Bram Van Leer Scheme are failing. The presence of shock wave, each numerical scheme requires more iteration number to achieve a convergent solution. The Modified Runge Kutta - Harten Yee scheme may represent an appropriate a numerical scheme for solving the flow problem with the presence of shock wave.

Efficient Computation of the Nonlinear Schrödinger Equation with Time-Dependent Coefficients

Motivated by the limited work performed on the development of computational techniques for solving the nonlinear Schrödinger equation with time-dependent coefficients, we develop a modified Runge–Kutta pair with improved periodicity and stability characteristics. Additionally, we develop a modified step size control algorithm, which increases the efficiency of our pair and all other pairs included in the numerical experiments. The numerical results on the nonlinear Schrödinger equation with a periodic solution verified the superiority of the new algorithm in terms of efficiency. The new method also presents a good behaviour of the maximum absolute error and the global norm in time, even after a high number of oscillations.

Exponentially fitted symmetric and symplectic DIRK methods for oscillatory Hamiltonian systems

The order conditions for modified Runge–Kutta methods are derived via the rooted trees. Symmetry and symplecticity conditions and exponential fitting conditions for modified diagonally implicit Runge–Kutta (DIRK) are considered. Three new exponentially fitted symmetric and symplectic diagonally implicit Runge–Kutta (EFSSDIRK) methods of respective second order and fourth order are constructed. Phase properties of the new methods are analyzed. The new EFSSDIRK methods are applied to several Hamiltonian problems and compared to the results obtained by the existing symplectic DIRK methods in the literature.

A Modified Runge–Kutta Method for Nonlinear Dynamical Systems With Conserved Quantities

In this paper, explicit Runge–Kutta methods are investigated for numerical solutions of nonlinear dynamical systems with conserved quantities. The concept, ε-preserving is introduced to describe the conserved quantities being approximately retained. Then, a modified version of explicit Runge–Kutta methods based on the optimization technique is presented. With respect to the computational effort, the modified Runge–Kutta method is superior to implicit numerical methods in the literature. The order of the modified Runge–Kutta method is the same as the standard Runge–Kutta method, but it is superior in preserving the conserved quantities to the standard one. Numerical experiments are provided to illustrate the effectiveness of the modified Runge–Kutta method.

Numerical modeling of exciton–polariton Bose–Einstein condensate in a microcavity

A novel, optimized numerical method of modeling of an exciton–polariton superfluid in a semiconductor microcavity was proposed. Exciton–polaritons are spin-carrying quasiparticles formed from photons strongly coupled to excitons. They possess unique properties, interesting from the point of view of fundamental research as well as numerous potential applications. However, their numerical modeling is challenging due to the structure of nonlinear differential equations describing their evolution. In this paper, we propose to solve the equations with a modified Runge–Kutta method of 4th order, further optimized for efficient computations. The algorithms were implemented in form of C++ programs fitted for parallel environments and utilizing vector instructions. The programs form the EPCGP suite which has been used for theoretical investigation of exciton–polaritons. Program summaryProgram title: EPCGPCatalogue identifier: AFBQ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AFBQ_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: BSD-3No. of lines in distributed program, including test data, etc.: 2157No. of bytes in distributed program, including test data, etc.: 498994Distribution format: tar.gzProgramming language: C++ with OpenMP extensions (main numerical program), Python (helper scripts).Computer: Modern PC (tested on AMD and Intel processors), HP BL2x220.Operating system: Unix/Linux and Windows.Has the code been vectorized or parallelized?: Yes (OpenMP)RAM: 200 MB for single runClassification: 7, 7.7.Nature of problem: An exciton–polariton superfluid is a novel, interesting physical system allowing investigation of high temperature Bose–Einstein condensation of exciton–polaritons-quasiparticles carrying spin. They have brought a lot of attention due to their unique properties and potential applications in polariton-based optoelectronic integrated circuits. This is an out-of-equilibrium quantum system confined within a semiconductor microcavity. It is described by a set of nonlinear differential equations similar in spirit to the Gross–Pitaevskii (GP) equation, but their unique properties do not allow standard GP solving frameworks to be utilized. Finding an accurate and efficient numerical algorithm as well as development of optimized numerical software is necessary for effective theoretical investigation of exciton–polaritons.Solution method: A Runge–Kutta method of 4th order was employed to solve the set of differential equations describing exciton–polariton superfluids. The method was fitted for the exciton–polariton equations and further optimized. The C++ programs utilize OpenMP extensions and vector operations in order to fully utilize the computer hardware.Running time: 6h for 100 ps evolution, depending on the values of parameters

New optimized symmetric and symplectic trigonometrically fitted RKN methods for second-order oscillatory differential equations

ABSTRACTSymmetric, symplectic and trigonometrically fitted Runge–Kutta–Nyström (SSTFRKN) methods for second-order differential equations with oscillatory solutions are investigated. Symmetry, symplecticity and trigonometric fitting conditions for modified Runge–Kutta–Nyström (RKN) methods are presented. Order conditions for modified RKN methods are derived via the special Nyström tree theory. Two explicit SSTFRKN methods with variable nodes are derived. The two new methods are zero-dissipative due to symplecticity. Their dispersion orders are analysed and their periodicity regions are obtained. The results of numerical experiments show the robustness and competence of the new SSTFRKN methods compared with some highly efficient codes in the recent literature.

Numerically efficient modified Runge–Kutta solver for fatigue crack growth analysis

We present a modified Runge–Kutta algorithm which yields a conservative estimate of the crack size for fatigue crack growth even for large integration step sizes. Conservative in this context means to overestimate the crack size. Commonly used algorithms (e.g. Euler, Runge–Kutta) usually underestimate the crack growth and only converge for small step sizes to an accurate value. As the presented algorithm overestimates the crack growth even for large and converges for small integration step sizes, it can be used for instance in Monte-Carlo based probabilistic fracture mechanics simulations, which might be otherwise computational impractical.

Modified Runge-Kutta Integration Algorithm for Improved Stability and Accuracy in Real Time Hybrid Simulation

Stability in Real Time Hybrid Simulation (RTHS) has been shown to be largely affected by system dynamics and associated phase lags. This lag typically originates in the physical components and considerable research has been conducted to compensate for it. Within the computational component of RTHS, different time integration algorithms are employed to achieve a more stable and accurate solution, mostly focusing on dissipating the high frequency content in the model. However, in RTHS, an inherent computational delay exists in the force measurement due to the sequential nature of communication between the numerical and experimental sub- structures. In this article, it is demonstrated that this computational delay affects performance and stability of closed loop RTHS even when no other delays or phase lags are present. This finding is validated through theoretical derivation and simulation results. A modified Runge-Kutta (MRK) integration algorithm is proposed to reduce the effect of computational delay.The MRK integration involves a three-stage computation: (1) the pseudo response is calculated using the delayed force measurement; (2) feedback force from the physical component for the next step is predicted using the pseudo response; and (3) the corrected structural response is then computed using the predicted feedback force. Both analytical and simulation results confirm that the MRK integration scheme is stable and accurate for a wide range time steps and is robust with respect to modeling error and nonlinearity in the experimental substructure. A moment-resisting frame is used as the experimental substructure in different cases of RTHS to validate the MRK integration method. This approach can also be adapted to other existing numerical integration schemes by applying the proposed three-stage computation process approach.