Heun's Method Research Articles

Comparison of Runge-Kutta Methods for Solving Nonlinear Equations

This research explores the numerical solutions of nonlinear ordinary differential equations using four distinct methods: Heun method, the Midpoint method, Ralston's method, and the fourth-order Runge-Kutta method (RK4). Given the complexity and prevalence of nonlinear equations in various scientific fields, effective numerical techniques are essential for obtaining accurate solutions. This research contributes to the understanding of numerical methods in solving complex differential equations, aiding practitioners in selecting the most appropriate approach for their needs.

A New Predictive Current Control With Reduced Current Tracking Error and Switching Frequency for Multilevel Inverters

The forward Euler's integration method is widely used to compute the trajectories of the control variables in conventional predictive current control (PCC) techniques. However, the computational error caused by the forward Euler's method proportionally increases with the sampling time. Hence, the current tracking ability of conventional PCC techniques deteriorates when operating with a large sampling time. Furthermore, the conventional PCC methods produce more switching transitions to reduce the current tracking error, thereby leading to a high switching frequency operation. To overcome these problems, Heun's integration method is proposed for the PCC in this article. The proposed method has predictor and corrector stages to minimize the computational error caused by the large sampling time during the prediction process of the control variables. Thereby, the proposed Heun's method-based PCC techniques produce fewer switching transitions to minimize the current tracking error. Consequently, this results in a decrease in switching frequency. The proposed PCC method is applied to a four-level multilevel inverter (4L-MLI). The discrete-time system models are developed using Heun's integration method to implement the proposed PCC. The working philosophy of the proposed method is demonstrated through the dSPACE/DS1103 controlled 4L-MLI laboratory prototype. The experimental performance of the proposed PCC is compared with that of the conventional PCC in terms of average switching frequency, current tracking error, current harmonic distortion, and transient response time.

Temperature dependent magnetic properties of crystallographically amorphous TbCo: An atomistic simulation study

Due to interesting properties such as perpendicular magnetic anisotropy (PMA), ferrimagnetism, etc., rare earth-transition metal(RE-TM) alloys attract considerable attention in ultrafast and terahertz spintronics applications. The magnetic properties of RE-TM alloys strongly depend on their composition and vary with temperature. Here, we model crystallographically amorphous ferrimagnetic TbCo using atomistic spin dynamics(ASD) simulations to study their temperature dependence on magnetic properties by varying the compositions. In this study, we investigate the temperature dependence of magnetisation, uniaxial anisotropy, and coercivity values of amorphous TbxCo100−x for the composition range Tb18Co82 to Tb36Co64. The monte carlo method is used to calculate the temperature dependence of magnetisation and calculate the magnetisation compensation temperature(TM) and curie temperature(Tc) values for different compositions. The temperature dependence of uniaxial anisotropy is calculated from the angular dependence of free energy curves calculated at different temperatures using constrained monte carlo methods. Further, we evaluate the temperature dependence of coercivity values by calculating the magnetic hysteresis at different temperatures using the LLG Heun method. The results show the magnetic properties of amorphous TbCo can be easily tuned by varying their composition and have a strong temperature dependence. These highly tunable magnetic properties make them promising candidates for spintronics applications.

SEIR Model Analysis for The Spread of Covid-19 Disease Post New Normal Using Euler Method and Matlab-Assisted Heun Method

In early 2020 Corona Virus Disease 2019 (COVID-19) was declared by WHO as a pandemic and the Government of Indonesia based on Presidential Decree Number 11 of 2020 concerning the Determination of Public Health Emergency of Corona Virus Disease 2019 (COVID-19). On this basis, this study aims to find out the solution to the numerical solution of the SEIR epidemic on the rate of Coronavirus Disease (COVID-19) by using the Euler method and the Heun method and to find out the design of a mathematics lesson plan in high school that is in accordance with the concept of the numerical method. In analyzing the spread of the Covid-19 disease after the new normal, a mathematical model of the Covid-19 disease was developed using the SEIR type with the stages of Literature Study, Mathematical Modeling, Model Analysis, Simulation and Drawing conclusions. The target to be achieved in this research is to produce mathematical modeling from the results of the analysis that carried out, reports can be confirmed and report results can be published in international journals. This research will produce TKT types of social humanities and education with the results of the status of R&D results are important and significant for supporting decisions and policies. Keywords : Model SEIR, EULER, HEUN, MATLAB. DOI: 10.7176/JEP/13-33-16 Publication date: November 30 th 2022

Analysis of power system angle stability based on Heun algorithm under gauss and jump excitation

In the process of promoting the dual-carbon goals, the “double-high” character of the power system is gradually becoming obvious and brings a lot of random factors, such as high proportion of new energy access and cut out power system, brings gauss excitation from continuous fluctuations and jump excitation from shock fluctuations, affecting the power system safety operation. Combining the effects of gauss excitation and jump excitation, described as excitation terms acting on gauss white noise and poisson white noise. Constructing the stochastic dynamic model with jump of power system, Heun method is constructed to solve the model, and the comprehensive influence of stochastic excitation and jump excitation under different intensities on stability analysis is analyzed quantitatively. The simulation results of one-machine infinite-bus (OMIB) system shows the correctness and rationality of the method constructed, and provides ideas for the analysis of power system stability under the combined influence of multiple types of random factors.

ANALYSIS OF THE SPRUCE BUDWORM MODEL USING THE HEUN METHOD AND THIRD-ORDER RUNGE-KUTTA

This study discusses the analysis of the Spruce Budworm model using numerical methods, namely the Heun method and the Third Order Runge-Kutta method. The purpose of this study is to determine the numerical results of the Heun method and the Third Order Runge-Kutta method on the cypress caterpillar model and to determine the comparison of errors from the two methods, namely the Heun method and the Third Order Runge-Kutta method in analyzing the Spruce Budworm model. The results of the study using the Heun method for the initial conditions at years, for , the result obtained is and . For the calculation result of the Spruce Budworm model using the third-order Runge-Kutta method, the result obtained is and . while the error of Caterpillar Density with the third-order Runge-Kutta method is bigger than the Heun method, while the error of Branch Surface Area and the error of Reserve Food error using the Heun method bigger than using the third-order Runge-Kutta method.

NUMERICAL SOLUTIONS OF COVID-19 SIRD MODEL IN NIGERIA

A mathematical model of the Susceptible, Infectious, Recovery and Death (SIRD) for the spread of the COVID-19 disease in Nigeria was considered in this paper. The model values and parameters were obtained from Nigeria Centre for Disease Control Coronavirus COVID-19. The model was solved using Heun's method and Runge-Kutta's method of orders four and five to obtain an approximated solution for the model with the help of the Matlab Program. The result obtained was illustrated in plots to show the progression of the disease in the various classes. The comparisons of the three numerical methods used correspond well with each other by showing the same behavior pattern

A new method for controlling an induction motor using a hybrid discretization model predictive field orientated control.

The dynamic performance of the Model Predictive Control (MPC) of an Induction Motor (IM) relies on the accuracy and computational efficiency of the Discretisation Technique (DT). If the discretisation process is inaccurate or slow approximation, the MPC will exhibit high torque ripple and lower load handling capabilities. Traditionally, Euler’s method is used to discretise the MPC, which merely relies on the predictor to yield a fast, but less accurate system approximation. In contrast, Heun’s method uses a combination of predictor and corrector at alternate sampling intervals to improve the discretisation accuracy; however, the controller response becomes slow due to increased computational intensity of the algorithm. In this study, a new Hybrid Discretisation Technique (HDT) for Model Predictive Field Oriented Control (MPFOC) for IM control systems is presented to achieve robust discretisation with improved accuracy. In the proposed approach, Euler’s method is used to discretise the system at the first nine samples, followed by the predictor-corrector at the tenth sampling interval, accomplishing the desired speed and accuracy of discretisation. This newly proposed HDT in MPFOC is verified with Processor-In-Loop (PIL) for a three-phase IM with bi-directional rotation under varying load conditions. The results indicate that the IM torque ripple is reduced by up to 20%, whereas, the load handling capability is increased by up to 10%. Moreover, the controller gives 20% and 23% improvement in rise time and settling time, respectively, under high loading conditions, as compared to traditional Euler and Heun methods.

Solution of the 1D KPZ Equation by Explicit Methods

The Kardar–Parisi-Zhang (KPZ) equation is examined using the recently published leapfrog–hopscotch (LH) method as well as the most standard forward time centered space (FTCS) scheme and the Heun method. The methods are verified by reproducing an analytical solution. The performance of each method is then compared by calculating the average and the maximum differences among the results and displaying the runtimes. Numerical tests show that due to the special symmetry in the time–space discretisation, the new LH method clearly outperforms the other two methods. In addition, we discuss the effect of different parameters on the solutions.

Effective residual convolutional neural network for Chagas disease parasite segmentation

Considered a neglected tropical pathology, Chagas disease is responsible for thousands of deaths per year and it is caused by the parasite Trypanosoma cruzi. Since many infected people can remain asymptomatic, a fast diagnosis is necessary for proper intervention. Parasite microscopic observation in blood samples is the gold standard method to diagnose Chagas disease in its initial phase; however, this is a time-consuming procedure, requires expert intervention, and there is currently no efficient method to automatically perform this task. Therefore, we propose an efficient residual convolutional neural network, named Res2Unet, to perform a semantic segmentation of Trypanosoma cruzi parasites, with an active contour loss and improved residual connections, whose design is based on Heun's method for solving ordinary differential equations. The model was trained on a dataset of 626 blood sample images and tested on a dataset of 207 images. Validation experiments report that our model achieved a Dice coefficient score of 0.84, a precision value of 0.85, and a recall value of 0.82, outperforming current state-of-the-art methods. Since Chagas disease is a severe and silent illness, our computational model may benefit health care providers to give a prompt diagnose for this worldwide affection.

Modeling the Effects of Chemotherapy and Immunotherapy on Tumor Growth.

Mathematical modeling has been used to simulate the interaction of chemotherapy and immunotherapy drugs intervention with the dynamics of tumor cells growth. This work studies the interaction of cells in the immune system, such as the natural killer, dendritic, and cytotoxic CD8+ T cells, with chemotherapy. Four different cases were considered in the simulation: no drug intervention, independent interventions (either chemotherapy or immunotherapy), and combined interventions of chemotherapy and immunotherapy. The system of ordinary differential equations was initially solved using the Runge-Kutta method and compared with two additional methods: the Explicit Euler and Heun's methods. Results showed that the combined intervention is more effective compared to the other cases. In addition, when compared with Runge-Kutta, the Heun's method presented a better accuracy than the Explicit Euler technique. The proposed mathematical model can be used as a tool to improve cancer treatments and targeted therapy.

On Adaptive Patankar Runge–Kutta methods

AbstractWe apply Patankar Runge–Kutta methods to y′ = M(y)y and focus on the case where M(y) is a graph Laplacian as the resulting scheme will preserve positivity and total mass. The second order Patankar Heun method is tested using four test problems (stiff and non‐stiff) cast into this form. The local error is estimated and the step size is chosen adaptively. Concerning accuracy and efficiency, the results are comparable to those obtained with a traditional L‐stable, second order Rosenbrock method.

Randomness Analysis With Runge Kutta Methods

Chaotic systems are widely used in encryption because of their sensitivity to initial conditions and parameters, high ergodicity, mixing properties, and highly complex structures. Various analyzes are available to understand whether a system is chaotic. The most used analyzes are time series analysis, phase portraits, Lyapunov exponents, and bifurcation diagrams. Modeling of chaotic systems is also possible with numerical analysis methods. These methods are; Houses, Heun, 4th and 5th degree Runge Kutta methods are the most common differential solution methods.

Numerical comparison by different methods (second order Runge Kutta methods, Heun method, fixed point method and Ralston method) to differential equations with initial condition

This manuscript contains a detailed comparison between numerical solution methods of ordinary differential equations, which start from the Taylor series method of order 2, stating that this series hinders calculations for higher order derivatives of functions of several variables, so that the Runge Kutta methods of order 2 are implemented, which achieve the required purpose avoiding the cumbersome calculations of higher order derivatives. In this document, different variants of the Runge-Kutta methods of order 2 will be exposed from an introduction and demonstration of the connection of these with the Taylor series of order 2, these methods are: the method of Heun, the method of midpoint and the Ralston method. It will be observed from the solution of test differential equations its respective error with respect to the analytical solution, obtaining an error index dictated by the mean square error EMC. Through this document we will know the best numerical approximation to the analytical solution of the different PVI (initial value problems) raised, also fixing a solution pattern for certain problems, that is, the appropriate method for each type of problem will be stipulated. It was observed that the Ralston method presented greater accuracy followed by the midpoint method and the Heun method, in the other PVI it is observed that the midpoint method yields the best numerical solution since it has a very low EMC and difficult to reach by the other methods.

A comprehensive theory on generalized BBKS schemes

In the past years several unconditionally positivity preserving numerical methods for production-destruction equations arising in a wide variety of real life applications were developed. The BBKS schemes suggested by Bruggemann et al. (2007) [4] and Broekhuizen et al. (2008) [3] modifications of Heun's method and conserve all linear invariants while still maintaining positivity independent of the time step size used. In this paper, we present a comprehensive generalization of these schemes as modifications of arbitrary first and second order Runge-Kutta methods with nonnegative parameters. This is achieved by the introduction of novel Patankar weights and sufficient and necessary conditions w.r.t. the Patankar weight denominators for first as well as second order consistency are proven. Furthermore, a convergence proof for the incorporated Newton scheme is given such that the efficiency compared to the original BBKS schemes, which make use of the bisection method is significantly improved. Numerical simulations are presented confirming the theoretical results concerning the validity, efficiency and accuracy of the new class of generalized BBKS schemes.

Improving time integration scheme for FET analysis of power system angle stability

This paper presents improved algorithm for numerical analysis of power system angle stability achieved by improvement of the time integration when forming a local system of equations for power system finite elements (FE). Previously developed local system of equations of power system angle stability has been obtained using the generalized trapezoidal rule (? - method). Improvement of accuracy was obtained by using Heun's method. Numerical solutions obtained using Heun?s method and using the generalized trapezoidal rule are compared to Electromagnetic Transients Program (EMTP). It has been shown that Heun?s method yields the results with much higher accuracy comparing to results obtained by generalized trapezoidal rule.

Attributes Weight Determination in Magdm Problems using Some Numerical Method Techniques

Here MAGDM problems are explored. Attribute weight data given are in the form of problems involving numerical method solutions. The decision information during this study is in the shape of Intuitionistic Fuzzy Numbers (IFNs) derived from Intuitionistic Fuzzy Sets (IFSs). The IFOWA operator and IFHA operator are used for collecting the combined matrix. The modified Euler-Cauchy method and Heun method are used to calculate the unidentified weights. Effectiveness of the projected method is established using Numerical Illustrations.

An efficient steady-state solver for microflows with high-order moment model

In Hu et al. (2016) [22], it has been successfully demonstrated that using lower-order moment model correction is a promising idea to accelerate the steady-state computation of high-order moment models of the Boltzmann equation. To develop the existing solver, the following aspects are studied in this paper. First, the finite volume method with linear reconstruction is employed for high-resolution spatial discretization of the resulting equations. Second, by introducing an appropriate parameter τ in the correction step, it is found that the performance of the solver can be improved significantly, i.e., more levels would be involved in the solver, which further accelerates the convergence of the method. Third, Heun's method is employed as the smoother in each level to enhance the robustness of the solver. Numerical experiments in microflows are carried out to demonstrate the efficiency and to investigate the behavior of the new solver. In addition, several order reduction strategies for the choice of the order sequence of the solver are tested, and the strategy ml−1=⌈ml/2⌉ is found to be most efficient.

Modelling and control of a wind turbine and farm

The Matlab/Simulink model of the Supergen (Sustainable Power Generation and Supply) Wind 5 MW exemplar wind turbine, which has been employed by a number of researchers at various institutions and Universities over the last decade, is reported. It is subsequently improved, especially in speed, to facilitate wind farm modelling, which usually involves duplicating wind turbine models. The improvement is achieved through various stages, including prewarping, discretisation using Heun's method in addition to Euler method, and conversion to C. Results are presented to demonstrate that improvement in speed is significant and that the resulting wind turbine model can be used for wind farm modelling more efficiently. It is important to highlight that improvement in speed is achieved without compromising the complexity of the turbine model; that is, each turbine included in a wind farm is neither simplified nor compromised. The use of the wind farm model for testing a wind farm controller that has recently been introduced is also demonstrated.

Numerical Solutions of Stochastic Differential Equations by using Heun's method

In this work, we study the numerical method for solving Stochastic differential equations. Because of the difficulty of finding analytical solutions for many of the Stochastic differential equations the Heun's method was used. Numerical simulations for different selected examples are implemented. And the difference between the numerical solution and the exact solution was also found.