Runge–Kutta Numerical Method Followed by Richardson’s Extrapolation for Efficient Ion Rejection Reassessment of a Novel Defect-Free Synthesized Nanofiltration Membrane

A defect-free, loose, and strong layer consisting of zirconium (Zr) nanoparticles (NPs) has been successfully established on a polyacrylonitrile (PAN) ultrafiltration substrate by an in-situ formation process. The resulting organic–inorganic nanofiltration (NF) membrane, NF-PANZr, has been accurately characterized not only with regard to its properties but also its structure by the atomic force microscopy, field emission scanning electron microscopy, and energy dispersive spectroscopy. A sophisticated computing model consisting of the Runge–Kutta method followed by Richardson extrapolation was applied in this investigation to solve the extended Nernst–Planck equations, which govern the solute particles’ transport across the active layer of NF-PANZr. A smart, adaptive step-size routine is chosen for this simple and robust method, also known as RK4 (fourth-order Runge–Kutta). The NF-PANZr membrane was less performant toward monovalent ions, and its rejection rate for multivalent ions reached 99.3%. The water flux of the NF-PANZr membrane was as high as 58 L · m−2 · h−1. Richardson’s extrapolation was then used to get a better approximation of Cl− and Mg2+ rejection, the relative errors were, respectively, 0.09% and 0.01% for Cl− and Mg2+. While waiting for the rise and expansion of machine learning in the prediction of rejection performance, we strongly recommend the development of better NF models and further validation of existing ones.

Application of discretization error estimators in stepped column buckling problems

In order to reduce the discretization error, in this paper, Richardson’s Extrapolation and Convergence Error Estimator were used to investigating the buckling problem convergence. The main objective was to verify the convergence order of the stepped column problem and to define a consistent moment of inertia at the point of variation of the cross-section. The variable of interest was the critical buckling load obtained by the Finite Difference Method. The convergent solution obtained errors less than 10-8, and this work showed that the best solution is not defined by excessive mesh refinement, but by the solution convergence analysis.

Numerical Analysis of Double Integral of Trigonometric Function Using Romberg Method

In general, solving the two-fold integral of trigonometric functions is not easy to do analytically. Therefore, we need a numerical method to get the solution. Numerical methods can only provide solutions that approach true value. Thus, a numerical solution is also called a close solution. However, we can determine the difference between the two (errors) as small as possible. Numerical settlement is done by consecutive estimates (iteration method). The numerical method used in this study is the Romberg method. Romberg's integration method is based on Richardson's extrapolation expansion, so that there is a calculation of the integration of functions in two estimating ways I (h1) and I (h2) resulting in an error order on the result of the completion increasing by two, so it needs to be reviewed briefly about how the accuracy of the method. The results of this study indicate that the level of accuracy of the Romberg method to the analytical method (exact) will give the same value, after being used in several simulations.

Comments on “heat transfer in a square porous cavity in presence of square solid block”

Purpose The purpose of this paper is to discuss the need to attend correctly to the accuracy and the manner in which the value of the streamfunction is determined when two or more impermeable boundaries are present. This is discussed within the context of the paper by Nandalur et al. (2019), which concerns the effect of a centrally located conducting square block on convection in a square sidewall-heated porous cavity. Detailed solutions are also presented which allow the streamfunction to take the natural value on the surface of the internal block. Design/methodology/approach Steady solutions are obtained using finite difference methods. Three different ways in which insulating boundary conditions are implemented are compared. Detailed attention is paid to the iterative convergence of the numerical scheme and to its overall accuracy. Error testing and Richardson’s extrapolation have been used to obtain very precise values of the Nusselt number. Findings The assumption that the streamfunction takes a zero value on the boundaries of both the cavity and the embedded block is shown to be incorrect. Application of the continuity-of-pressure requirement shows that the block and the outer boundary take different constant values. Research limitations/implications The Darcy–Rayleigh number is restricted to values at or below 200; larger values require a finer grid. Originality/value This paper serves as a warning that one cannot assume that the streamfunction will always take a zero value on all impermeable surfaces when two or more are present. A systematic approach to accuracy is described and recommended.

Algorithm 1 and Algorithm 2 for Determining the Number pi (π)

Archimedes used the perimeter of inscribed and circumscribed regular polygons to obtain lower and upper bounds of π. Starting with two regular hexagons he doubled their sides from 6 to 12, 24, 48, and 96. Using the perimeters of 96 side regular polygons, Archimedes showed that 3+10/71<π<3+1/7 and his method can be realized as a recurrence formula called the Borchardt-Pfaff-Schwab algorithm. Heinrich Dörrie modified this algorithm to produce better approximations to π than these based on Archimedes’ scheme. Lower bounds generated by his modified algorithm are the same as from the method discovered earlier by cardinal Nicolaus Cusanus (XV century), and again re-discovered two hundred years later by Willebrord Snell (XVII century). Knowledge of Taylor series of the functions used in these methods allows to develop new algorithms. Realizing Richardson’s extrapolation, it is possible to increase the accuracy of the constructed methods by eliminating some terms in their series. Two new methods are presented. An approximation of squaring the circle with high accuracy is proposed.

Corrected discrete approximations for the conditional and unconditional distributions of the continuous scan statistic

The (conditional or unconditional) distribution of the continuous scan statistic in a one-dimensional Poisson process may be approximated by that of a discrete analogue via time discretization (to be referred to as the discrete approximation). Using a change of measure argument, we derive the first-order term of the discrete approximation which involves some functionals of the Poisson process. Richardson's extrapolation is then applied to yield a corrected (second-order) approximation. Numerical results are presented to compare various approximations.