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Unsteady cavitating flow often contains vapor structures with a wide range of different length scales, from micro-bubbles to large cavities, which issues a big challenge to precisely investigate its evolution mechanism by computational fluid dynamics (CFD) method. The present work reviews the development of simulation methods for cavitation, especially the emerging Euler-Lagrange approach. Additionally, the progress of the numerical investigation of hot and vital issues is discussed, including cavitation inception, cloud cavitation inner structure and its formation mechanism, cavitation erosion, and cavitation noise. It is indicated that the Euler-Lagrange method can determine cavitation inception point better. For cloud cavitation, the Euler-Lagrange method can reveal the source of microbubbles and their distribution law inside the shedding cloud. This method also has advantages and great potential in assessing cloud cavitation-induced erosion and noise. With the ever-growing demands of cavitation simulation accuracy in basic research and engineering applications, how to improve the Euler-Lagrange method’s stability and applicability is still an open problem. To further promote the application of this advanced CFD simulation technology in cavitation research, some key issues are to be solved and feasible suggestions are put forward for further work.

This paper aims to investigate the motion patterns of magnetically interacting gears with a few teeth and given initial angular velocities. Based on Ampere’s molecular current hypothesis, an equivalent current model is utilized to solve the magnetic field generated by cylindrical permanent magnets. Comsol software is employed to simulate the distribution of the magnetic field, demonstrating that the magnetic field calculated by the equivalent current model is in good agreement with the actual situation. Vector analysis, field theory, and the symmetry of the magnetic field distribution are leveraged to simplify the calculations, thereby reducing the runtime of the program. The Euler method is adopted to numerically simulate the motion of the gears. Finally, a comparison with experiments shows that the motion patterns in the experiments are essentially consistent with those in the numerical simulations. It is concluded that the ratio of magnetic field intensity to resistance torque, initial magnetic potential energy, and initial angular velocity positively correlate with the motion duration; the ratio of magnetic field intensity to resistance torque and initial magnetic potential energy positively correlate with the intensity of motion; when the initial angular velocity is excessively high, the other gear exhibits slight vibrations in its original position, and only when the initial velocity decreases to a certain value do the two gears interact significantly.

In the gradient descent method, one often focus on the convergence of the sequence generated by the algorithm, but less often on the deviation of these points from the solutions of the original continuous-time differential equation (gradient flow). This also happens when discretizing other ordinary differential equations. In the case of a discretization by explicit Euler’s method with a constant step h, we provide here sufficient conditions, in terms of strong monotonicity and co-coercivity, for the deviation between discrete and continuous solutions to tend asymptotically towards zero. This analysis could shed new light on some applications of the gradient descent algorithm.

This article represents a numerical analysis for a stochastic dengue epidemic model with incubation period of virus. We discuss the comparison of solutions between the stochastic dengue model and a deterministic dengue model. In this paper, we have shown that the stochastic dengue epidemic model is more realistic as compared to the deterministic dengue epidemic model. The effect of threshold number R1\\(R_{1}\\) holds in the stochastic dengue epidemic model. If R1<1\\(R_{1} <1\\), then situation helps us to control the disease while R1>1\\(R_{1} >1\\) shows the persistence of disease in population. Unfortunately, the numerical methods like Euler–Maruyama, stochastic Euler, and stochastic Runge–Kutta do not work for large time step sizes. The proposed framework of stochastic nonstandard finite difference scheme (SNSFD) is independent of step size and preserves all the dynamical properties like positivity, boundedness, and dynamical consistency.

Accurate flux observation is crucial for the high-performance control of induction motors (IMs). Implementing a full-order flux observer algorithm in digital controllers requires discretizing the continuous-domain full-order flux observer. However, the errors introduced by discretization increase with rising rotor speed. In the field-weakening region, inappropriate discretization methods can lead to significant flux estimation errors, severely affecting the performance of model predictive control-based induction motors and potentially causing system instability. To enhance the convergence speed and stability of the observer and reduce discretization errors in the field-weakening region, this paper designs a feedback gain matrix suitable for high-speed field-weakening regions and conducts a study and summary of commonly used discretization methods. Discrete full-order flux observer models based on the forward Euler method, improved Euler method, and third-order Runge–Kutta method are designed. The discretization error, stability, and model complexity of the observers using these three discretization methods in the field-weakening region are analyzed. The experimental results demonstrate that the improved Euler method can achieve high discretization accuracy with relatively low computational complexity, making it a suitable discretization approach for full-order flux observers.

In this paper, we mainly consider the oscillation of numerical solutions for a nonlinear delay differential equation which is generalized from a delay Lotka-Volterra type single species population growth model. By studying the corresponding difference scheme of the equation discretized by θ-method, forward Euler method and backward Euler method, some sufficient conditions under which the numerical solutions oscillate are obtained. Furthermore, we prove that the positive non-oscillatory numerical solutions tend to the equilibrium of the original differential equation. Finally, some numerical experiments are given to verify the theoretical results.

On the one hand, the explicit Euler scheme fails to converge strongly to the exact solution of a stochastic differential equation (SDE) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient. On the other hand, the implicit Euler scheme is known to converge strongly to the exact solution of such an SDE. Implementations of the implicit Euler scheme, however, require additional computational effort. In this article we therefore propose an explicit and easily implementable numerical method for such an SDE and show that this method converges strongly with the standard order one-half to the exact solution of the SDE. Simulations reveal that this explicit strongly convergent numerical scheme is considerably faster than the implicit Euler scheme.

The gas-suspended proppant is composed of conventional proppant particles and hydrophobic aerophilic coating. The surface coating changes the surface wettability of the proppant and increases the adhesion of the proppant surface to the gas, changing the traditional liquid-carried proppant into a gas-carried proppant. The sand carrying capacity of low concentration polymer fracturing fluid can be greatly improved by using air-suspended proppant carrying technology. At present, there is no numerical simulation method aiming at the migration law of gas-suspended proppant. Based on the principle of equivalent substitution, the density of gas and particles is set as the calculated equivalent density during simulation. Similarly, the diameter of gas-suspended proppant is set as the calculated equivalent diameter during simulation. Then computational fluid dynamics software was used to simulate the Euler-Euler method. It is found that Euler-Euler method can simulate the placement of gas-suspended proppant in fractures. Compared with conventional proppant, gas-suspended proppant is not easy to settle during migration, and the settling sand bed still has fluidity and can be re-suspended. When gas suspension proppant density increases, the proppant sedimentation trend increased. Increase the injection rate of proppant, and the shape of long and high sand dunes will become shorter and lower. As the sand rate increases, the height of the sand bank rapidly increases.

This research aims to solve a comprehensive one-dimensional model of drug release from cardiovascular stents in which the drug binding is saturable and reversible. We used the Lagrange collocation method for space dimension and the modified Euler method for time discretization. The existence and uniqueness of the solution, are provided. The consistency, stability, and convergence analysis of the proposed scheme are provided, to show that numerical simulations are valid. Numerical results accurate enough and efficient just by using fewer mesh.

A method recently advanced as the conformable Euler method (CEM) for the finite difference discretization of fractional initial value problem Dtαyt = ft;yt, yt0 = y0, a≤t≤b, and used to describe hyperchaos in a financial market model, is shown to be valid only for α=1. The property of the conformable fractional derivative (CFD) used to show this limitation of the method is used, together with the integer definition of the derivative, to derive a modified conformable Euler method for the initial value problem considered. A method of constructing generalized derivatives from the solution of the non-integer relaxation equation is used to motivate an alternate definition of the CFD and justify alternative generalizations of the Euler method to the CFD. The conformable relaxation equation is used in numerical experiments to assess the performance of the CEM in comparison to that of the alternative methods.