Euler Method Research Articles

Viscosity-modulated clustering of heated bidispersed particles in a turbulent gas

Clustering of externally and evenly heated particles is enhanced by the increased viscosity of heated fluid in the vicinity of these clusters – a phenomenon known as viscous capturing (VC). Herein we study, via direct numerical simulations of decaying turbulence, the effect of temperature-driven viscosity on clustering with different particle loading densities. We employ a two-way momentum and energy coupling, and gas viscosity is modelled by a power law to understand the role of the increased drag and particle back-reaction force on the clustering intensity. For the continuum and dispersed phases, Eulerian and Lagrangian point particle schemes have been used, neglecting inter-particle collisions. We found that the enhanced viscosity-driven clustering is a strong function of particle loading density, as the increase in particle number density enables the formation of large uneven clusters before heating, which is the main condition for VC to take effect. Higher number density should result in greater turbulence modulation and negate local temperature-based viscous effects leading to VC. However, due to higher local particle number density in the clusters and interphase heat transfer, increased drag force prevails in such cases and delivers excessive clustering. By sampling conditionally the particle velocity and temperature inside the clusters, it is found that the thermodynamic and kinematic properties of the particles in the clusters are highly correlated, and this correlation increases with the particle loading density. Therefore, based on the particle number density, temperature-based viscosity can enhance considerably the clustering of heated particles and alter the effect of particles on the underlying turbulence.

Anisotropic constitutive modeling with the considerations of two strategies in describing the distorted hardening of AA6061 under tension-dominated loading conditions

This paper investigates the anisotropic yielding and distorted hardening behaviors of AA6061-T6. Yoon 2014 model was used to describe the anisotropic plastic behaviors of material and two strategies (i.e. different equivalent plastic strain intervals at two levels) in describing the distorted hardening were used. At the integration point level, a program utilizing the forward Euler method and regression mapping algorithms has been developed. This program accepts strain as input and provides corresponding stress as output. Results found that the prediction ability of constitutive model strongly depend on the strain interval of parameters identification. Excessive interval might result in the unexpected fluctuations of stress-strain curve. Such fluctuations primarily come from the deviation of identified parameters and interpolation data. The former deviation is found caused by the changes in initial value during identifying and the latter mainly because the interpolation data departures from the actual values. In addition, a detailed analysis about parameters effects is also conducted for in-depth understanding of parameters identification.

A hybrid computational scheme for singularly perturbed Burgers’-Huxley equation

This paper aims to construct and analyze a hybrid computational method for the nonlinear singularly perturbed Burgers’-Huxley equation. The presence of the perturbation parameter and non-linearity in the considered problem makes it difficult to solve the problem analytically and using classical numerical techniques on uniform step sizes as ε goes small. To elucidate such limitations, one can rely on non-classical numerical techniques. In this paper, one such parameter uniform numerical method is designed for the considered problem. The method is constructed:•Firstly, the non-linear terms are linearized via Newton-Raphson-Kantorovich technique. The linearized problem is discretized by the implicit Euler method in the temporal direction.•Secondly, the obtained equation is solved by employing the hybrid computational method comprised of the cubic spline in tension method in the inner layer region and the midpoint upwind method in the outer layer region on a piecewise uniform Shishkin mesh.•Finally, an error analysis of the method is done and observed that the proposed method is parameter uniform convergent with the order of convergence O(τ+N−2ln3N). Three examples are presented and the results are compared to some existing schemes in the literature to demonstrate the reliability of the proposed scheme.

Non-negativity-preserving and maximum-principle-satisfying finite difference methods for Fisher’s equation with delay

Little attention has been devoted to the numerical studies on maximum-principle-satisfying FDMs for Fisher’s equation with delay. Monotone difference schemes can preserve the maximum principle of the continuous problem. However, it is difficult to develop monotone difference schemes for Fisher’s equation with delay because of delay term. The main novelties of this study are to develop the maximum-principle-satisfying FDMs for it by using cut-off technique to adjust the numerical solutions obtained applying non-negativity-preserving FDMs. Firstly, by using a new weighted difference formula with parameter θ, new numerical formula and explicit Euler method to discrete the diffusion term, delay term and temporal variable, respectively, a class of new explicit non-negativity-preserving FDMs are established for one-dimensional problem. Then, by applying cut-off technique to adjust their numerical solutions, a kind of new explicit maximum-principle-satisfying FDMs are developed. Secondly, based on previous work, by using implicit Euler method for the approximation to the temporal variable, an implicit non-negativity-preserving FDM is designed. Likewise, by applying cut-off technique to adjust the obtained numerical solutions, an implicit maximum-principle-satisfying FDM is devised. By convergent analyses, cut-off techniques do not reduce convergent rates. Thirdly, the extensions of our methods to two-dimensional problems are discussed. Finally, numerical results confirm the correctness of theoretical findings and the efficiency of our methods for long-term simulations.

Parallel computation to bidimensional heat equation using MPI/CUDA and FFTW package

In this study, we present a fast algorithm for the numerical solution of the heat equation. The heat equation models the heat diffusion over time and through a given region. We engage a finite difference method to solve this equation numerically. The performance of its parallel implementation is considered using Message Passing Interface (MPI), Compute Unified Device Architecture (CUDA), and time schemes, such as Forward Euler (FE) and Runge-Kutta (RK) methods. The originality of this study is research on parallel implementations of the fourth-order Runge-Kutta method (RK4) for sparse matrices on Graphics Processing Unit (GPU) architecture. The supreme proprietary framework for GPU computing is CUDA, provided by NVIDIA. We will show three metrics through this parallelization to compare the computing performance: time-to-solution, speed-up, and performance. The spectral method is investigated by utilizing the FFTW software library, based on the computation of the fast Fourier transforms (FFT) in parallel and distributed memory architectures. Our CUDA-based FFT, named CUFFT, is performed in platforms, which is a highly optimized FFTW implementation. We will give numerical tests to reveal that this method is up-and-coming for solving the heat equation. The final result demonstrates that CUDA has a significant advantage and performance since the computational cost is tiny compared with the MPI implementation. This vital performance gain is also achieved through careful attention of managing memory communication and access.

Weak approximation schemes for SDEs with super-linearly growing coefficients

We propose a new class of weak approximation schemes for stochastic differential equations with coefficients of suplinearly growth. Both the modified weak Euler schemes and the drift-implicit weak Euler scheme are studied. Under certain non-globally Lipschitz conditions, the proposed schemes are proved to have first-order convergence in the weak sense. Numerical experiments are included to confirm the theoretical results.

Solvability and Ulam-Hyers stability analysis for nonlinear piecewise fractional cancer dynamic systems

We examine a nonlinear dynamical model that depicts the interaction between cancerous cells and an oncolytic virus. For best modelling the disease, we use the Caputo fractional derivative in piecewise approaches. By employing piecemeal techniques, we treat a compartment in the body that contains infectious and non-infectious cells. More precisely, the solvability and Ulam-Hyers (U-H) stability results are considered using standard concepts. Further, to support our investigation with numerical results, we apply the Euler method to develop an approximation solution. It connected with numerous graphical representations of the system using various arbitrary ordering and varying values of the isolation parameters. Here we remark that the multi-step behavior that certain problems exhibit, is one of important issues naturally. This paper introduces the idea of piecewise derivative with the goal of modeling real-world issues that follow multiples processes. With the help of the used approach, we investigate the cancer disease model and its transmission dynamical behavior with crossover effect.

On the complex version of the Cahn–Hilliard–Oono type equation for long interactions phase separation

Abstract This paper focuses on the complex version of the Cahn-Hilliard-Oono equation with Neumann boundary conditions, which is used to capture long-range nonlocal interactions in the phase separation process. The first part of the paper establishes the well-posedness of the corresponding stationary problem associated with the equation. Subsequently, a numerical model is constructed using a finite element discretization in space and a backward Euler scheme in time. We demonstrate the existence of a unique solution to the stationary problem and obtain error estimates for the numerical solution. This, in turn, serves as proof of the convergence of the semi-discrete scheme to the continuous problem. Finally, we establish the convergence of the fully discrete problem to the semi-discrete formulation.

Spectral element discretization of the time-dependent Stokes problem with nonstandard boundary conditions

This work deals with the spectral element discretization of the time-dependent Stokes problem in two- and three-dimensional domains. The boundary condition is defined on the normal component of the velocity and the tangential components of the vorticity. The discretization related to the time variable is processed by a Backward Euler method. We prove through a detailed numerical analysis the well-posedness of the full discrete problem.

Active Control of Quasi-Zero-Stiffness Vibration Isolator with Variable Load

Quasi-zero-stiffness (QZS) isolator has great application potential in the field of low-frequency vibration isolation due to its high-static and low-dynamic (HSLD) nonlinear stiffness characteristic, but it is precisely this characteristic that makes it very sensitive to load changes. Once the load changes, causing it to deviate from the equilibrium position, it no longer qualifies zero-stiffness characteristic, and the vibration isolation capacity will be significantly decrease. To make the QZS isolator possess variable load capacity and be more suitable for engineering practice, an active QZS vibration isolator based on electromagnetic actuator is designed in this paper, which eliminates the influence of load changes through the active force of the electromagnetic actuator. First, the dynamic equation of the isolator is established by Newton–Euler method, and the dynamic characteristic of the isolator under standard load and load variations are analyzed through improved incremental harmonic balance (IHB) method based on discrete Fourier transform (DFT). Next, the improved particle swarm optimization (PSO) algorithm are employed to optimize the Proportion Integration Differentiation (PID) controller parameters. Then, the vibration isolation performance of QZS isolator in controlled and uncontrolled and linear systems in the same control state are compared in frequency domain and time domain, respectively. Finally, the performance of active QZS isolator under load variation is discussed. The results indicate that the isolation performance of the QZS isolator under active control is significantly better than in uncontrolled conditions and the controlled linear system. When the load changes, real-time compensation through the actuator output control force can also enable the QZS isolator to achieve a better vibration isolation performance.

Effects of Drilling Number and Distribution on Fracture Using the Pulse Plasma on Tight Sand Reservoir

Summary The permeability of unconventional reservoirs is extremely low, resulting in their drainage area being limited to tens of feet. Therefore, researchers have developed an effective stimulation technology that can be used in combination with conventional hydraulic fracturing, namely, pulsed plasma fracturing technology. Pulsed plasma fracturing technology is an efficient and environmentally friendly auxiliary hydraulic fracturing stimulation technology. However, most existing studies have focused only on the effect of pulsed plasma fracturing on single wells, ignoring the effect of the number and distribution of wells drilled on pulsed plasma fracturing. In this paper, pulsed plasma fracturing is studied by a self-built pulsed plasma experimental platform and nonlinear finite element software. First, the generation and propagation mechanism of shock wave, fracture type, and stress field analysis of rock mass in pulsed plasma fracturing technology are discussed. The double-well experiment was carried out by using the experimental platform, and the fracture law of fractures under different wellhead distribution conditions was obtained. In addition, a multiwell mathematical model is established by using the combination of the Euler method and Lagrange method to simulate the interaction between fluid and solid, that is, arbitrary Lagrangian Eulerian (ALE) multimaterial fluid-solid coupling method and the influence of drilling times and wellhead distribution on pulsed plasma fracturing is discussed. Stress analysis shows that the rock is mainly affected by ground stress, liquid column pressure, and shock wave pressure. The experimental results show that the discharge voltage is positively correlated with the shock wave pressure on the rock. The distribution of different wellheads affects the distribution and length of fractures. The double-well experiment makes the fractures easier to fracture. The simulation results show that the fracture length in the connection direction of the two wells is longer, and the fracture length in the vertical direction is shorter. This shows that the number and distribution of drilling affect the initiation and propagation of fractures.

A mathematical analysis to the approximate weak solution of the Smagorinsky Model for different flow regimes

This study delves into the numerical approximation of non-stationary Navier-Stokes equations within turbulent regimes, employing the Smagorinsky Model (SM). By treating the model as inherently discrete, we implement a semi-implicit time discretization using the Euler method. This approach includes comprehensive stability analyses, applicable to a spectrum of flow regimes, and an exploration of the asymptotic energy balance dynamics during fluid movements. The primary contribution of this study is found in its methodical approach to the numerical approximation of non-stationary Navier-Stokes equations within turbulent regimes using the Smagorinsky Model (SM). The adoption of a semi-implicit time discretization with the Euler method, coupled with a meticulous analysis of energy balance, establishes a robust foundation adaptable to diverse flow conditions.

Controlling a generalized Fokker–Planck equation via inputs with nonlocal action

This paper concerns an optimal control problem (P) related to a generalized Fokker–Planck equation. Basic properties of the solutions to the generalized FP equation are derived via a semigroup approach in the space H−1(Rd). Problem (P) is proven to be deeply related to a stochastic optimal control problem (PS) for a McKean–Vlasov equation. The existence of an optimal control is obtained for the deterministic problem (P). The existence of an optimal control is established for an approximating optimal control problem (Ph) related to a backward Euler approximation of the generalized FP equation (with a constant discretization step h). One proves under additional hypotheses that “(Ph) converges to (P)” in a certain sense. First order necessary conditions for (Ph) are derived as well.

Periodic Event-Triggered Model Predictive Control for Networked Nonlinear Uncertain Systems With Disturbances.

This article investigates the event-triggered model predictive control (MPC) problem for a class of networked nonlinear uncertain systems subject to time-varying disturbances. Different from the traditional MPC, the proposed periodic event-triggered MPC (PETMPC) method does not generate new control sequence unless a predesigned periodic event-triggering mechanism (PETM) is violated. First, a generalized proportional-integral observer (GPIO) is developed to estimate the unknown state and disturbance information by using the sampled-data output of controlled system. Then, the disturbance predictions for future finite steps are obtained based on forward Euler method. After that, with the help of prediction model, the optimal control sequence, including the future finite step predicted control inputs, is generated and dexterously exploited during the interevent interval by storing it in a buffer installed between the control sequence generator and actuator, thereby leading to the further reduction of signal transmission number and the frequency of control sequence computations. Through a rigorous stability analysis, it can be proved that the closed-loop hybrid control system is globally bounded stable under the nominal PETMPC law. Finally, numerical simulations are conducted to substantiate the feasibility and superiority of the proposed PETMPC method.

The Impact of Frame Transformations on Power System EMT Simulation

This article investigates the impact of frame transformations on the accuracy of numerical discretization in power system transient and stability studies. As analysed, frame transformations influence the convergence of the numerical discretization. Specifically, for an explicit discretization method (e.g., forward Euler method), the stability of the original system is best preserved in the frame where the system eigenvalue is closer to the origin of the complex plane, e.g., in the stationary frame for inductors and capacitors, and in the synchronous frame for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dq-</i> frame controllers of inverters. Simulation results are given to validate the theoretical analysis.

Numerical Solutions of Initial Value Problem by using Modified Euler's Method and Taylor's Method

Numerical Solutions of Initial Value Problem by using Modified Euler's Method and Taylor's Method

Existence and uniqueness of solutions for stochastic differential equations with locally one-sided Lipschitz condition

&lt;p&gt;This paper investigated stochastic differential equations (SDEs) with locally one-sided Lipschitz coefficients. Apart from the local one-sided Lipschitz condition, a more general condition was introduced to replace the monotone condition. Then, in terms of Euler's polygonal line method, the existence and uniqueness of solutions for SDEs was established. In the meanwhile, the $ p $th moment boundedness of solutions was also provided.&lt;/p&gt;

Convergence analysis of Euler and BDF2 grad-div stabilization methods for the time-dependent penetrative convection model

&lt;abstract&gt;&lt;p&gt;Based on the grad-div stabilization method, the first-order backward Euler and second-order BDF2 finite element schemes were studied for the approximations of the time-dependent penetrative convection equations. The proposed schemes are both unconditionally stable. We proved the error bounds of the velocity and temperature in which the constants are independent of inverse powers of the viscosity and thermal conductivity coefficients when the Taylor-Hood element and $ P_2 $ element are used in finite element discretizations. Finally, numerical experiments with high Reynolds numbers were shown to confirm the theoretical results.&lt;/p&gt;&lt;/abstract&gt;

A novel Lagrangian–Eulerian weighted-least squares scheme coupled with other stable techniques for multi-physical fluid flow around complex obstacle

In this paper, a stable novel meshless coupled method is proposed to simulate the non-isothermal magnetohydrodynamics (MHD) flow problems (multi-physics quantities) inside a lid-driven cavity around complex obstacle. The proposed method is mainly motivated by a Lagrangian–Eulerian (L–E) weighted-least squares (WLS) scheme combined with a stream function-vorticity (SFV) and other stable techniques, and it is further to investigate the non-isothermal MHD flow around an airfoil obstacle at large Hartmann (Ha) or Reynolds (Re) number, for the first time. In the present meshless coupled approach (named L–E WLS–SFV), the traditional MHD equations are derived as another form with an SFV method under divergence-free constraint, which can avoid the tedious treatment of pressure on complex irregular obstacle. Then, a stable L–E WLS coupled algorithm is proposed to approximate the space derivatives of multi-physical quantities (velocity, magnetic, temperature, etc.), in which a corrected particle shifting technique is employed to improve the tensile instability among Lagrangian particles moving inside the domain and a second-order upwind scheme is adopted to stabilize large Re number problem in Eulerian fixed nodes near the boundary. Several benchmarks are simulated to show the numerical accuracy and convergence rates of the proposed WLS scheme for MHD flow at different parameters. Subsequently, the case of the non-isothermal MHD flow around a square obstacle under large parameters is simulated by the proposed L–E WLS–SFV method and compared with other numerical results to demonstrate the validity and capacity of the proposed method for multi-physical flow and the necessity of imposing the above two stable techniques. Finally, the case of non-isothermal MHD flow around the circular or airfoil obstacle is numerically investigated, and the important effects of the Hartmann, Rayleigh, and Reynolds numbers on the multi-physical quantities (stream function, vorticity, temperature, and magnetic field) are discussed. The advantages of the proposed method for the muti-physical flow around irregular obstacles are also exemplified. All the numerical results show that the proposed L–E WLS–SVF method is robust and accurate to simulate the multi-physical fluid flow around complex obstacles.

Optimization and control in rubella transmission dynamics: A boundedness-preserving numerical model with vaccination

Rubella is an infectious disease that can spread globally. It spreads in the subtropics as well as the tropics. Although it is commonly thought to be a non-fatal condition, there are several circumstances in which it can be fatal. Pregnant women infected with the Rubella virus face a high risk of fetal development. The main goal of this study is to look into a model for the spread of Rubella while considering a vaccination campaign as a control measure. The positivity and boundedness of the re-infection rubella transmission and vaccination model are investigated. The local and global stability of the equilibrium points is examined. The sensitivity of parameters is also investigated. Three different techniques forward Euler, RK-4, and non-standard finite difference (NSFD) method are developed for the numerical solution, and their simulation results are examined. Among these three, the NSFD method is superior due to its convergence and positive behavior for all step size values. For some values of the step size, the other two techniques failed to produce positivity and convergent solutions. Analytically, the proposed model's convergence, positivity, boundedness, and consistency are investigated. Finally, the impact of the Rubella vaccine on infected populations has been examined, revealing that vaccination is one of the most effective ways to prevent rubella transmission.