Finite Element Iterative Methods for the 3D Steady Navier–Stokes Equations

In this work, a finite element (FE) method is discussed for the 3D steady Navier–Stokes equations by using the finite element pair Xh×Mh. The method consists of transmitting the finite element solution (uh,ph) of the 3D steady Navier–Stokes equations into the finite element solution pairs (uhn,phn) based on the finite element space pair Xh×Mh of the 3D steady linearized Navier–Stokes equations by using the Stokes, Newton and Oseen iterative methods, where the finite element space pair Xh×Mh satisfies the discrete inf-sup condition in a 3D domain Ω. Here, we present the weak formulations of the FE method for solving the 3D steady Stokes, Newton and Oseen iterative equations, provide the existence and uniqueness of the FE solution (uhn,phn) of the 3D steady Stokes, Newton and Oseen iterative equations, and deduce the convergence with respect to (σ,h) of the FE solution (uhn,phn) to the exact solution (u,p) of the 3D steady Navier–Stokes equations in the H1−L2 norm. Finally, we also give the convergence order with respect to (σ,h) of the FE velocity uhn to the exact velocity u of the 3D steady Navier–Stokes equations in the L2 norm.

On the Transmission of COVID-19 and Its Prevention and Control Management

The spread of an epidemic is a typical public emergency and also one of the major problems that humans need to tackle in the 21st century. Therefore, the research on the spread, prevention, and control of epidemics is quite an essential task. This paper first briefly described and analyzed the development of COVID-19 and then introduced the basic epidemic models and idealized the population in the epidemic area by dividing them into four categories (Classes S, E, I, and R). After that, it set the relevant parameters of the basic SEIR model and the modified one and worked out the relevant differential equations and iterative equations. According to the feature of the epidemic situation and the changes in the number of contacts in different units of time, the epidemic data were substituted into the iterative equations for data fitting with an R Package. Then, analysis was performed on the epidemiological features such as the transmission time and epidemic peak and the epidemic trend was evaluated. Finally, sensitivity analysis was conducted on the parameters (government control and recovery rate), and the results showed that measures such as government restrictions on travel (reducing the contacts between virus carriers and susceptible persons) can effectively control the scale of the outbreak.

Experimental and Theoretical Validation of One Diode and Three Parameters–Based PV Models

The present paper defines and assesses a new simplified method to represent the photovoltaic (PV) modules’ electrical behavior, based on the commonly used one diode and three parameters (1D + 3P) model, addressing two main objectives. The first one is to quantify and assess, at different operating conditions, the PV modules electrical behavior estimations’ accuracy provided by the well-known 1D + 3P, through a comparison based on experimental and theoretical results. The second one concerns the performance assessment of the 1D + 3P model’s suggested approximation, aiming at simplifying the mathematics instead of solving complex iterative equations, which hinges on higher computational time to obtain accurate results. Hence, experimental and theoretical data were considered, aiming at performing a thorough comparison with more than 17,000 PV modules being assessed, which was achieved by using both the California Energy Commission (CEC) database and PVsyst software. The findings show that the already known 1D + 3P model delivers satisfactory power output estimations for crystalline silicon modules and high irradiance conditions. However, its performance worsens when considering Low Irradiance and thin-film technology. In comparison with the original model, accurate results were obtained with the new simplified suggested 1D + 3P for all irradiance conditions and technologies assessed, thus proving its validity and capability of circumventing the aforementioned challenges.

Revisiting the Light Time Correction in Gravimetric Missions Like GRACE and GRACE Follow-On

<p>The satellite gravimetry missions GRACE (Gravity Recovery and Climate Experiment) and GRACE Follow-On provide the global and monthly gravity field for almost 17 years, which plays an irreplaceable role in understanding the mass transport of the Earth system. The key observation is the biased inter-satellite range, which is measured primarily by a K-Band Ranging system (KBR) in GRACE and GRACE Follow-On. The GRACE Follow-On satellites are additionally equipped with a Laser Ranging Interferometer (LRI), which provides measurements with lower noise compared to the KBR. However, the measured biased range which is directly measured by the inter-satellite ranging systems differs from the instantaneous biased range which is usually required for gravity field recovery. The difference is called the Light Time Correction (LTC) and arises from the non-zero travel time of electromagnetic waves between the spacecraft. We re-analyzed the LTC calculation from first principles considering general relativistic effects and state-of-the-art models of Earth’s potential field, and different types of orbital data. By analyzing the iterative equations in the LTC calculation of KBR and LRI, a novel analytical expression method is obtained to avoid the numerical limitation of the classical method. The dependency of the LTC on geopotential models and on the parameterization is further studied, and afterwards the results are compared against the LTC provided in the official datasets of GRACE and GRACE Follow-On. It is shown that the new approach has significantly lower noise, well below the instrument noise of current instruments, especially relevant for the LRI, and even if used with kinematic orbit products. This allows calculating the LTC accurate enough even for the next generation of gravimetric missions.</p>

Exact Solution of a Linear Difference Equation in a Finite Number of Steps

An exact solution of a linear difference equation in a finite number of steps has been obtained. This refutes the conventional wisdom that a simple iterative method for solving a system of linear algebraic equations is approximate. The nilpotency of the iteration matrix is the necessary and sufficient condition for getting an exact solution. The examples of iterative equations providing an exact solution to the simplest algebraic system are presented.

A relativistic formulation of the de la Peña-Cetto stochastic quantum mechanics

A covariant generalization of a non-relativistic stochastic quantum mechanics introduced by de la Peña and Cetto is formulated. The analysis is done in space-time and avoids the use of a non-covariant time evolution parameter in order to search for Lorentz invariance. The covariant form of the set of iterative equations for the joint coordinate and momentum distribution function Q(x; p) is derived and expanded in power series of the coupling of the particle with the stochastic forces. Then, particular solutions of the zeroth order in the charge of the iterative equations for Q(x; p) are considered. For them, it follows that the space-time probability density ρ(x) and the function S(x) which gradient defines the mean value of the momentum at the space time point x, define a complex function ψ(x) which exactly satisfies the Klein-Gordon (KG) equation. These results for the zeroth order solution reproduce the ones formerly and independently derived in the literature. It is alsoargued that when the KG solution is either of positive or negative energy, the total number of particles conserves in the random motion. Other solutions for the joint distribution function in lowest order, satisfying the positive condition are also presented here. The are consistent with the assumed lack of stochastic forces implied by the zeroth order equations. It is also argued that such joint distributions, after considering the action of the stochastic forces, might furnish an explanation of the quantum mechanical properties, as associated to ensembles of particles in which the vacuum makes such particles behave in a similar way as Couder’s droplets moving over oscillating liquid surfaces. Some remarks on the solutions of the positive joint distribution problem proposed in the Olavos’s analysis are also presented.