Comparison between Euler, Heun, Runge-Kutta and Adams-Bashforth-Moulton integration methods in the particle dynamic simulation

Euler's integration methods are frequently used for numerical integration as well as for real-time implementation of linear systems. However, when the integrated system is either undamped or stiff, Euler's explicit integration becomes unstable, regardless of the forcing input. In this work, it is shown that this instability can be avoided by a judicious selection of the state variables. Instead of the generalized coordinates and velocities, it is proposed to define state variables based on the method of variation of parameters. It is proven that the variation-of-parameters-based state variables yield a bounded numerical error for undamped and stiff systems, provided that the forcing inputs are bounded. The analysis is performed for both deterministic and stochastic inputs. In the stochastic case, the numerically calculated covariance matrix entries diverge when using the generalized coordinates and velocities, but remain bounded when implementing the variation-of-parameters-based approach. The newly developed formalism is illustrated by a number of examples of practical interest, showing that the variation-of-parameters-based approach is also more computationally efficient than the standard approach.

Aim: In this paper, analysis of ventricular arrhythmias are made with respect to the Calcium (Ca2+) ion channel dysfunction (generating improper electrical activity). Many cases can make arrhythmias and most of them are related to generation or conduction of Action Potential (AP) in cardiac myocardium. Materials and method: Human ventricular cell based on the model of the human endocardial cell by Ten Tusscher (TT). The TT model data is modified based on the experimental data of Han, describing the properties of Ca2+ currents and its channel dynamics in human ventricular cells. Euler integration method is used to analyse the human ventricular model for different channel failure conditions in the same group of 50 samples. Results: Our research findings focus with respect to normal and deviant Ca2+ conductance (GCaL). The normal GCaL 0.000175nS and deviant GCaL increase like (10%=0.000218nS, 25%=0.000182nS, 50%=0.000262nS and 100%=0.000350nS) having the normal AP average value ranges between 26.0mV to -74.0mV and 12.0mV to -88.0mV for 10% GCaL, 18.0mV to -78.0mV for 25% GCaL, 18.0mV to -78.0mV for 50% GCaL and 21.0mV to -75.0mV for 100% GCaL deviants. Similarly, deviant GCaL decrease like (10%=0.000158nS, 25%=0.000131nS, 50%=0.000088nS and 100%=0.000001nS) having the deviant AP mean values ranges between 10.0mV to -90.0mV for 10% GCaL, 7.0mV to -92.0mV for 25% GCaL, -9.0mV to -96.0mV for 50% GCaL and -51.0mV to 100.0mV for 100% GCaL. Simultaneously its membrane Ca2+ currents are having significant variations. Conclusion: The results show clearly for the affirmation for Excitation and Coupling (EC) failures. EC failures lead to a systole phase that is more prolonged, that in turns to produce QT syndrome and hypertrophic cardiomyopathy.

Computational based approach to analyze Sodium (Na + ) channels effects in Purkinje fiber cells for action potential Phase 1 using Euler integration method - a simulation study

Predicting students’ course grades is an essential element in teaching. This paper used decision tree generation rules to study the prediction of students’ ideological and political course grades. Firstly, ID3 and C4.5 algorithms were briefly introduced; then, an improved C4.5 algorithm with higher computational efficiency was put forward. The formula of the C4.5 algorithm was optimized using theories such as the Taylor series. Finally, experiments were performed on the UCI dataset and students’ ideological and political course datasets. The results suggested that the average classification accuracy and computation time of the improved C4.5 algorithm was 79.37% and 74.1 ms, respectively, on the UCI dataset, which was better than the traditional C4.5 algorithm. Then, the experiment predicting students’ course grades demonstrated that the average quiz grade and the number of video views had the greatest impact on the final grades. The prediction accuracy of the improved C4.5 algorithm reached 93.46%, and the average computation time was 54.8 ms, which was 19.17% less than the C4.5 algorithm. The experimental results verify the effectiveness of the generation rule of the improved C4.5 algorithm in predicting students’ ideological and political course grades. This algorithm can be applied in the actual grade prediction.

The paper discusses the problem of integrating the equations of state observers associated with direct field orientation (DFO) of motor drives and studies the influence of the discretization method used on the accuracy of integration. In a typical implementation, discrete-time integration is done using Euler's discretization method (forward rectangular rule) - the method is simple and integration is accurate when the drive operates at low and medium speed. However, as the frequency increases, the integration becomes inaccurate because the Euler approximation starts losing more and more area from under the curve. Theoretically, the problem could be alleviated by increasing the sampling frequency; however, this cannot always be done. Another idea would be to adopt a more accurate (but more computationally intensive) integration method, for example, trapezoidal integration (Tustin method). The paper shows that, at high frequency, under ideal conditions, trapezoidal integration performs better than the Euler method. In a real implementation, however, conditions are non-ideal since the measured signals bring dc offsets and imperfections into the terms to be integrated - as a result, pure integration must be replaced with quasi-low pass filtering. Under these conditions, the paper compares the Euler, Tustin and backward rectangular methods from the point of view of integration accuracy. The implications related to direct field orientation of motor drives are studied by considering a full-order observer for the PMSM - this is discretized using the three methods considered and the results are compared. At high frequency, neither integration method gives perfect results; the Euler method yields a waveform that leads the expected one while the backward rectangular method yield a waveforms that lags it. The paper finds that, surprisingly, when quasi-low pass filtering is used, the Tustin method is not significantly more accurate than the other ones - the waveform obtained lags the expected one by an angle comparable with the lead angle of the Euler method. It is shown that the integration accuracy depends on the frequency, sampling time, filter bandwidth and on the integration method used. Accurate high frequency drive DFO control would require correction of the magnitude/phase of the estimates.

The turning process with varying spindle speed is investigated. The well-known single degree of freedom turning model is presented and the governing delay-differential equation with time varying delay is analyzed. Three different numerical techniques are used to solve the governing equation: (1) direct Euler simulation with linear interpolation of the delayed term, (2) Taylor expansion of the time delay variation combined with Euler integration and (3) semi-discretization method. The results of the three method are compared. Stability charts are constructed, and some improvements in the process stability is shown, especially for low spindle speed domains.

Cai and Selman [CS] proposed the following definition for measuring average computation time: A time function t is T on average over a distribution μ if, for all $ n \geq 1 $ , $ \sum_{|x|\geq n}T^{-1}(t(x))|x|^{-1}\mu (x) \leq W_n $ , where $ W_n = \mu (\{ x : |x| \geq n\} ) $ . This definition results in a modification of Levin's notion of average time [L]. The effect of the modification is to control the rate of convergence of the expressions that define average computation time. With this modification, they proved a hierarchy theorem for average-time complexity that is as tight as the Hartmanis—Stearns [HS] hierarchy theorem for worst-case deterministic time. They also proved that under a fairly reasonable condition on distributions, called condition W, a distributional problem is solvable in average polynomial time under the modification exactly when it is solvable in average polynomial time under Levin's definition. Various notions of reductions, as defined by Levin [L] and others, play a central role in the study of average-case complexity. However, the class of distributional problems that are solvable in average polynomial time under the modification is not closed under the standard reductions. In particular, we prove that there is a distributional problem that is not solvable in average polynomial time under the modification but is reducible, by the identity function, to a distributional problem that is, and whose distribution even satisfies condition W.

On numerical integration of the Hodgkin and Huxley equations for a membrane action potential

We have developed an accurate and efficient method for molecular dynamics simulations of charged particles confined by planar dielectric interfaces. The algorithm combines the image-charge method for near field with the harmonic surface mapping, which converts the contribution of infinite far-field charges into a finite number of charges on an auxiliary spherical surface. We approximate the electrostatic potential of far-field charges via spherical harmonic expansion and determine the coefficients by fitting the Dirichlet-to-Neumann boundary condition, which only requires the potential within the simulation cell. Instead of performing the direct evaluation of spherical harmonic series expansion, we use Green's second identity to transform the series expansion into a spherical integral, which can be accurately represented by discrete charges on the sphere. Therefore, the fast multipole method can be readily employed to sum over all charges within and on the sphere, achieving truly linear O(N) complexity. Our algorithm can be applied to a broad range of charged complex fluids under dielectric confinement.

This paper deals with prediction of wear in spur gears, which has been calculated using an approach in which the wear depth on a tooth surface is integrated over time using the Euler integration method. Variations in pressure distribution over the surface have been considered, as well as changes in curvature and surface profile of the teeth as they wear. The paper focuses on the use of an appropriate wear model for simulating wear using calculated conditions between high‐performance slow‐running lubricated spur gears. A numerical model for prediction of surface interaction of spur gears developed earlier by the authors has been applied as the basis for the simulation of wear using different wear models. Owing to the nature of the wear models, methods of calculating the contact temperature have been investigated for the specific case of interacting gear teeth. It was found that a linear wear model, when compared with a more complex oxidation model and with a model based on adsorption of lubricant molecules, satisfactorily predicts wear. It was also found that it is the geometry of the gears that causes sliding velocity to be the dominant wear‐inducing factor, and so determines the general nature of the wear behaviour.

The bond graph theory provides a firm and complete strategy for making mathematical models and are used in this work to obtain a good relation between connectivity, causality and model fidelity in distributed systems. By distributing a system more computational power is available which makes it\npossible to increase the model fidelity in large systems without increasing the time to solve the total system. Also, more complex models with causality switching properties may be used for simplifying the connectivity problem between distributed models and for representing changing dynamics that also affects the model causality.\nStability of distributed systems are dependent on both solver stability and dynamical stability, when neglecting the stability results based on cascaded systems with certain passivity properties. For linear distributed systems solver with fixed step size solvers a stability criterion involving the system dynamics, local solver time step and global synchronization time step can be formulated. In this work a stability criterion for linear distributed systems solved with the Euler integration method will be derived and a hybrid causality model, representing a small power plant, will be used to test the stability criterion.

<p>A new simple approach inspired by MacAyeal (1979) to explain the time-asymmetric ‘saw-toothed’ shape and 100,000-year quasi-period of glacial-interglacial cycles since the Middle Pleistocene Transition, is presented. Using a simple model with fast-slow dynamics, the global ice volume is taken to be a function of two independently varying parameters, the solar insolation and ‘alpha’, a secondary control parameter, the study of which is the focus this research. The steady state of the model is a partially folded surface in three-dimensional space where insolation, ‘alpha’, and global ice volume are orthogonal axes. The pleated surface allows for the gradual increase and sudden decrease in ice volume that is observed in the paleoclimate record. To derive a time series of global ice volume, the Euler integration method is used, producing a time series which replicates the ‘saw-toothed’ pattern of glacial cycles in the late Pleistocene. The second control parameter, ‘alpha’, is proposed to be related to internal dynamics of the climate system, such as ice sheet dynamics.</p><p> </p><p>Reference</p><p>D. R.  MacAyeal, ‘A Catastrophe Model of the Paleoclimate Record’ , Journal of Glaciology , Volume 24 , Issue 90 , 1979 , pp. 245 – 257.</p>

In this paper, a new approach for the realization of 1D discrete cosine transform (1D-DCT) is presented. This approach is based on a new formulation of an odd prime-length DCT which uses two half-length cyclic convolutions with the same form which can be computed in parallel. Using this approach, a new efficient systolic array with outstanding performance in structural regularity, hardware cost of the PE's, average computation time, and I/O cost has been obtained, which is well suited for VLSI realization. The average computation time has been reduced to one half and the throughput has been doubled, as compared with a previous design. It possesses also a much lower control complexity, a simpler interconnection structure, and a simpler hardware structure of the PE's, having thus a shorter cycle time. Moreover, it owns all other outstanding features of the systolic array previously proposed.

In this paper, what we believe to be, a new combined algorithm of artificial bee colony and slime mould algorithm (CABCSMA) and a differential evolution (DE) algorithm using target-to-best variation strategy are proposed to process the data based on Planck's radiation law and the mathematical model of reference temperature. The material model with 6 different emissivity trends is simulated. Simulation results show that the average relative error of CABCSMA algorithm is less than 0.68%, and the average calculation time is 0.44s. The average relative error of DE algorithm is less than 0.43%, and the average calculation time is only 0.06s. The two algorithms were applied to the temperature test of silicon carbide sample, tungsten material and rocket engine nozzle. The experimental results show that the relative error of silicon carbide experimental temperature is less than 0.41% and 0.28%, and the relative error of tungsten material experimental temperature is less than 0.31% and 0.3%. The relative errors of rocket engine nozzle temperature experiments are within 0.68% and 0.52%, respectively. The results show that these two algorithms are expected to be applied in practical measurement scenarios.

<p class="Abstract">This research aims to develop a numerical method that can accurately estimate the relationship between cosmic energy (E) and the expansion rate of the universe (H), taking into account the complex interactions between ordinary matter, dark matter, and dark energy. Numerical approaches based on Euler, Runge-Kutta, and Adams-Bashforth integration methods will be refined to evaluate the correlation. The limitation of this study is to a flat universe (k = 0 geometry), but it has the potential to be extended to other geometries. This effective numerical method can revolutionize cosmology by allowing accurate testing of cosmological theories and improving predictive capabilities. This study not only deepens our understanding of the behavior of the universe, but also opens up opportunities for further exploration. While there has been research on the Friedmann equation and the evolution of the universe, this study fills the gap by comparing three numerical methods, promising a more comprehensive and accurate analysis. This research demonstrates significant advances in cosmological methodology, with the potential to change the cosmological paradigm through efficient numerical approaches. By improving the understanding of cosmic energy and the expansion rate of the universe, this research not only contributes to the current knowledge of cosmology, but also paves the way for impactful follow-up research in this field.</p>