Accurate Numerical Solutions Research Articles

Evaporation of arrays of single-component drops on hydrophobic and hydrophilic surfaces

Abstract The paper focuses on modelling the evaporation of arrays of single-component identical sessile droplets, a phenomenon encountered in various applications. An analytical approach is used and developed to treat ordered and scattered sessile drop arrays of any numerosity. An approximate analytical solution for the vapour field and evaporation fluxes from general arrays of sessile drops on any type of substrate (hydrophobic or hydrophilic) is found by a proper superposition of analytical solutions of single drop fields. Comparison with accurate numerical solutions and available experimental results validates the approach. An analysis of ordered and scattered square arrays from 4 to 49 drops on substrates with contact angles from 30° to 150°, is reported and used to study the dependence of the evaporation screening coefficients on array patterns.

Application of fluid dynamics in modeling the spatial spread of infectious diseases with low mortality rate: A study using MUSCL scheme

Abstract This study presents a comprehensive mathematical framework that applies fluid dynamics to model the spatial spread of infectious diseases with low mortality rates. By treating susceptible, infected, and treated population densities as fluids governed by a system of partial differential equations, the study simulates the epidemic’s spatial dynamics. The Monotone Upwind Scheme for Conservation Laws is employed to enhance the accuracy of numerical solutions, providing a high-resolution approach for capturing disease transmission patterns. The model’s analogy between fluid flow and epidemic propagation reveals critical insights into how diseases disperse geographically, influenced by factors like human mobility and environmental conditions. Numerical simulations show that the model can predict the evolution of infection and treatment population densities over time, offering practical applications for public health strategies. Sensitivity analysis of the reproduction number highlights the influence of key epidemiological parameters, guiding the development of more efficient disease control measures. This work contributes a novel perspective to spatial epidemiology by integrating principles of fluid dynamics, aiding in the design of targeted interventions for controlling disease outbreaks.

Monolithic Newton‐Multigrid Finite Element Methods for the Simulation of Thixoviscoplastic Flows

ABSTRACTIn this paper, we shall be concerned with the development, application, and numerical analysis of the monolithic Newton‐Multigrid finite element method (FEM) to simulate thixoviscoplastic (TVP) flows. We demonstrate the importance of robustness and efficiency of Newton‐Multigrid FEM solver for obtaining accurate solutions. To put our work in proper perspective w.r.t. the delicate challenge of obtaining accurate numerical solutions for TVP flow problems, we content our investigation to TVP quasi‐Newtonian modeling approach with an extensive analysis on lid‐driven cavity flows, and expose the impact of thixotropic scale in 4:1 contraction configuration application. fldauth.cls class file for setting papers for the International Journal for Numerical Methods in Fluids. Copyright 2010 John Wiley & Sons Ltd.In this paper, we shall be concerned with the development, application, and numerical analysis of the monolithic Newton‐Multigrid finite element method (FEM) to simulate thixoviscoplastic (TVP) flows. We demonstrate the importance of robustness and efficiency of Newton‐Multigrid FEM solver for obtaining accurate solutions. To put our work in proper perspective w.r.t. the delicate challenge of obtaining accurate numerical solutions for TVP flow problems, we restrict our investigation to TVP quasi‐Newtonian modeling approach and lid‐driven cavity flows.

Logarithmic Bernstein functions for fractional Rosenau-Hyman equation with the Caputo-Hadamard derivative

In this study, the Caputo-Hadamard derivative is fittingly used to define a fractional form of the Rosenau-Hyman equation. To solve this equation, the orthonormal logarithmic Bernstein functions (BFs) are created as a suitable basis for handling this type of derivative. The primary benefit of these functions lies in the ease of computing their Hadamard fractional integral and derivative. These logarithmic functions, combined with the orthonormal Bernstein polynomials (BPs), are simultaneously employed to develop a hybrid strategy for solving the aforementioned equation. More precisely, the orthonormal logarithmic BFs are utilized to approximate the solution in the temporal domain and the orthonormal BPs are employed in the spatial domain. In addition, a matrix is extracted for the Hadamard integral of the orthonormal logarithmic BFs due to the implementation of the presented method. The effectiveness of the established scheme in finding accurate numerical solutions is evaluated through the resolution of three examples.

Highly accurate computation of chronopotentiograms for a charge transfer followed by homogeneous dimerization or disproportionation reactions

A theoretical model of constant current chronopotentiometry is considered, for a reversible charge transfer followed by an irreversible homogeneous dimerization or disproportionation reactions at a planar electrode. The model is expressed by a system of two partial differential reaction–diffusion equations, one or both being nonlinear. This system decomposes into two independent initial boundary value problems, which allows one to obtain semi-analytical series solutions of the model. Detailed derivations and analysis of the series are presented. Highly accurate numerical reference solutions are also obtained. Hybrid algorithms are devised for computing electrode potential-time responses. The algorithms combine the series solution for small time, with asymptotic approximants for large time, and fitted polynomials for intermediate time. The algorithms are efficient and highly accurate: the relative error modulus does not exceed ca. 10−19−10−18, except for the small neighbourhoods of the transition time, and of the time where the potential response passes through zero. A set of C++ routines for these calculations is made available.

The High-Order ADI Difference Method and Extrapolation Method for Solving the Two-Dimensional Nonlinear Parabolic Evolution Equations

In this paper, the numerical solution for two-dimensional nonlinear parabolic equations is studied using an alternating-direction implicit (ADI) Crank–Nicolson (CN) difference scheme. Firstly, we use the CN format in the time direction, and then use the CN format in the space direction before discretizing the second-order center difference quotient. In addition, we strictly prove that the proposed ADI difference scheme has unique solvability and is unconditionally stable and convergent. The extrapolation method is further applied to improve the numerical solution accuracy. Finally, two numerical examples are given to verify our theoretical results.

Investigating neural networks with groundwater flow equation loss

Physics-Informed Neural Networks (PINNs) are considered a powerful tool for solving partial differential equations (PDEs), particularly for the groundwater flow (GF) problem. In this paper, we investigate how the deep learning (DL) architecture, within the PINN framework, is connected to the ability to compute a more or less accurate numerical GF solution, so the link ‘PINN architecture - numerical performance’ is explored. Specifically, this paper explores the effect of various DL components, such as different activation functions and neural network structures, on the computational framework. Through numerical results and on the basis of some theoretical foundations of PINNs, this research aims to improve the explicability of PINNs to resolve, in this case, the one-dimensional GF equation. Moreover, our problem involves source terms described by a Dirac delta function, providing insights into the role of DL architecture in solving complex PDEs.

A Fast Time-Discontinuous Peridynamic Method for Stress Wave Propagation Problems in Saturated Porous Media

When employing numerical methods to simulate the behavior of saturated porous media under impact load, the Gibbs phenomenon often occurs, significantly compromising the accuracy of numerical calculations. To improve the accuracy of numerical solutions, this study introduces a fast time-discontinuous peridynamic (TDPD) method for simulating the propagation of fluid pressure and solid stress waves within saturated porous media. In this method, the classical u–p and u–U models are reformulated from spatial differential equations into integral equations to facilitate the simulation of fracture problems. Subsequently, the basic field variables are independently interpolated in the temporal domain, with the introduction of jump terms representing the discontinuities of variables between adjacent time steps. Then an integral weak form in the temporal domain of the spatially discrete governing equations is constructed and the basic formula of the TDPD is derived. Additionally, an efficient approximation method that directly uses the internal force to calculate its time derivative is further proposed. These characteristics ensure that the TDPD can effectively and efficiently capture the inherent sharp gradient features of wave propagation in solid phase and fluid while controlling spurious numerical oscillations. Several representative numerical examples demonstrate that the TDPD can obtain more accurate results compared with conventional peridynamic solution schemes. Moreover, the TDPD can also be viewed as a novel time integration technique, holding substantial potential for high-precision numerical solutions of hyperbolic equations in diverse physical contexts.

A comparative analysis of CFD methodologies to predict the performance of Francis turbine

Abstract A Francis turbine’s flow characteristics strongly depend on the operating conditions. Accurately predicting the flow field within a turbine unit is challenging due to complex flow phenomena, such as separation around the turbine blades, pressure variations within the runner, and swirling flows in the draft tube. Various CFD methodologies can be employed to predict turbine performance. The present work evaluates the reliability of different CFD simulation methodologies in predicting turbine performance, focusing on the sensitivity of different turbulence models, grid types, and modeling approaches for accurate numerical solutions. Simulations are performed on a high-head Francis turbine under various operating conditions, including part load, Best Efficiency Point (BEP), and high load conditions, each representing a different turbine flow field. The simulation results are compared with experimental data from model tests to confirm the reliability of the numerical simulations for Francis turbines. The results show that the SST k-ω model with a hexahedral mesh (HH) is best suited for overall turbine performance prediction.

Enhanced Adomian Decomposition Method for Accurate Numerical Solutions of PDE Systems

This research addresses critical challenges in numerical solutions, which are vital for various engineering and physical fields. The Modified Adomian Decomposition Method (MADM) is proposed as a novel approach for solving linear and nonlinear partial differential equations (PDEs). MADM builds upon the Adomian Decomposition Method (ADM) by incorporating a new integral operator that significantly improves convergence rates and accuracy. Numerical examples demonstrate the effectiveness of MADM in handling complex nonlinear PDEs. Compared to traditional ADM, MADM consistently achieves more accurate and rapidly converging solutions. This enhancement is attributed to the novel integral operator, which addresses the limitations of ADM for intricate nonlinear problems. The paper outlines the application of MADM, its solution procedure, and its effectiveness through numerical examples. Comparisons with standard ADM solutions and exact solutions validate MADM's accuracy and superiority. The results suggest that MADM is a promising tool for expanding the applicability of Adomian methods in the field of solving PDEs.

High Energy Density Radiative Transfer in the Diffusion Regime with Fourier Neural Operators

Radiative heat transfer is a fundamental process in high energy density physics and inertial fusion. Accurately predicting the behavior of Marshak waves across a wide range of material properties and drive conditions is crucial for design and analysis of these systems. Conventional numerical solvers and analytical approximations often face challenges in terms of accuracy and computational efficiency. In this work, we propose a novel approach to model Marshak waves using Fourier Neural Operators (FNO). We develop two FNO-based models: (1) a base model that learns the mapping between the drive condition and material properties to a solution approximation based on the widely used analytic model by Hammer & Rosen (2003), and (2) a model that corrects the inaccuracies of the analytic approximation by learning the mapping to a more accurate numerical solution. Our results demonstrate the strong generalization capabilities of the FNOs and show significant improvements in prediction accuracy compared to the base analytic model.

Total Positivity and Accurate Computations Related to q-Abel Polynomials

The attainment of accurate numerical solutions of ill-conditioned linear algebraic problems involving totally positive matrices has been gathering considerable attention among researchers over the last years. In parallel, the interest of q-calculus has been steadily growing in the literature. In this work the q-analogue of the Abel polynomial basis is studied. The total positivity of the matrix of change of basis between monomial and q-Abel bases is characterized, providing its bidiagonal factorization. Moreover, well-known high relative accuracy results of Vandermonde matrices corresponding to increasing positive nodes are extended to the decreasing negative case. This further allows to solve with high relative accuracy several algebraic problems concerning collocation, Wronskian and Gramian matrices of q-Abel polynomials. Finally, a series of numerical tests support the presented theoretical results and illustrate the goodness of the method where standard approaches fail to deliver accurate solutions.

Utilizing numerical techniques for modeling radiating Casson nanofluid flow with thermophoretic phenomenon on a heated stretching surface

This work aims to use mathematical modeling to investigate the mechanics of heat and mass transport in dissipative Casson nanofluid flows over a linear rough sheet. This study considers various elements such as thermal radiation, magnetic fields, heat generation, the varied properties of porous media, and the thermophoretic impact. Besides that, it looks into the variations in viscosity, diffusivity, and thermal conductivity, as well as the energy dissipation from viscous internal friction and fluid temperature modifications. The method involves coming up with and changing the boundary layer equations into a group of linked nonlinear ordinary differential equations that use variables that do not have any dimensions. The foundation of the solution strategy for these equations is the Hermite collocation method (HCM), which is renowned for its precision and adaptability. It offers an organized approach to solving the complex differential equations, allowing for accurate numerical solutions. The use of graphical representations ensures thorough data analysis and clarity, while also providing insightful information about the computed outcomes. The code validation method uses numerical comparisons with recent research to confirm the algorithm’s accuracy and dependability, as well as its resilience in comparison to the existing literature. Important conclusions from the study show that thermal boundary layers and nanofluid velocity decrease with increases in the porosity parameter, slip parameter, and magnetic field parameter.

A PNP ion channel deep learning solver with local neural network and finite element input data

Abstract This paper presents a deep learning method for solving an improved one-dimensional Poisson–Nernst–Planck ion channel (PNPic) model, called the PNPic deep learning solver. The solver combines a novel local neural network, adapted from the neural network with local converging inputs, with an efficient PNPic finite element solver, developed in this work. In particular, the local neural network is extended to handle the complexities of the PNPic model—a system of nonlinear convection–diffusion and elliptic equations with multiple subdomains connected by interface conditions. The PNPic finite element solver efficiently generates input and reference datasets for fast training the local neural network, as well as input datasets for quickly predicting PNPic solutions with high accuracy for a family of PNPic models. Initial numerical tests, involving perturbations of model parameters and interface locations, demonstrate that the PNPic deep learning solver can generate highly accurate numerical solutions.

DAPLSR: Data Augmentation Partial Least Squares Regression Model via Manifold Optimization

Traditional Partial Least Squares Regression (PLSR) models frequently underperform when handling data characterized by uneven categories. To address the issue, this paper proposes a Data Augmentation Partial Least Squares Regression (DAPLSR) model via manifold optimization. The DAPLSR model introduces the Synthetic Minority Over-sampling Technique (SMOTE) to increase the number of samples and utilizes the Value Difference Metric (VDM) to select the nearest neighbor samples that closely resemble the original samples for generating synthetic samples. In solving the model, in order to obtain a more accurate numerical solution for PLSR, this paper proposes a manifold optimization method that uses the geometric properties of the constraint space to improve model degradation and optimization. Comprehensive experiments show that the proposed DAPLSR model achieves superior classification performance and outstanding evaluation metrics on various datasets, significantly outperforming existing methods.

The dependency of the analytical and numerical solution on the $\varepsilon$ parameter in hyperbolic and pseudo-hyperbolic problems with inverse coefficients

The aim of this study is to analyze the behavior of $\varepsilon$ on the solution of an inverse coefficient nonlinear pseudo-hyperbolic equation $w_{tt}-\varepsilon w_{xxtt}-\varepsilon w_{xx}=\theta (t)f(x,t,w)$ with periodic boundary conditions. We also consider the inverse coefficient problem $w_{tt}-w_{xx}=\theta (t)f(x,t,w).$ The solution function of nonlinear pseudo-hyperbolic equation is found to be convergent to the solution function of nonlinear hyperbolic equation, when $ \varepsilon \rightarrow 0$ is proved. The Fourier method was used to illustrate the theoretically relation between the inverse problems while the Finite Difference Method was used numerically. In order to get more accurate numerical solution higher precision schemes have been applied in implicit finite difference equation. The cases where $\varepsilon =0$ and $\varepsilon \neq 0$ have been solved analytically and numerically, and compared each other.

High-Order Models for Gas Transport in Multiscale Coal Rock.

The coal reservoirs exhibit great heterogeneity and strong anisotropy in multiscale pore/fracture structures. Developing highly accurate multiscale models for real-time prediction of microgaseous flow in complicated porous media with pronounced contrast in transport coefficients is crucial but not yet available, which is time-consuming, expensive, and even computationally impossible. In this study, a multiscale approximate solution of the gas flow and pressure field is derived in this paper for predicting coalbed methane (CBM) transport in macro-microscopic two-scale porous media of typical coal rocks in Guizhou Province, China, and the detailed finite element algorithm is established. The multiscale approximate solutions are constructed from the first- and second-order auxiliary cell functions and low- or high-order derivatives of the homogenized pressure field, which can accurately approximate the solution of the original problem to a certain extent. The numerical accuracy and validity of the multiscale approximate solutions under various working conditions are analyzed in detail by comparing and verifying the results with the direct numerical simulation results of fine-mesh high-precision finite elements. The results show that the homogenized approximate solution can only roughly capture the characteristics of the macroscopic pressure evolution, the first-order approximate solution can partially reproduce the macro-microscopic pressure oscillation phenomenon, and the second-order approximate solution can accurately predict the pressure profile and its evolution law with time in porous media with macro-micro configuration under various complex conditions. The second- and high-order two-scale approximate solutions constructed in this paper exhibit high numerical accuracy and excellent computational efficiency. We also develop multiscale approximate solutions for predicting microgaseous flow at a low reservoir pressure where the slippage effects are very pronounced. Potential applications of our high-order models to the coal reservoirs with a hierarchical matrix and fractures are discussed. The developed multiscale models exhibit excellent applicability to investigating the crossflow effects and coupling mechanisms between the matrix and cleats or fractures with multiple configurations in the CBM and shale gas reservoirs, which is crucial for the effective implementation of hydraulic fracturing technology. This study provides a fundamental understanding of gas transport in multiscale matrix-fracture networks, which helps to improve the optimization design of hydraulic fracturing in tight reservoirs.

Investigation of the dynamics of transverse oscillations of a vertical rod under gravity, friction, and thermal expansion

The paper considers the mathematical formulation of the problem of transverse oscillations of a vertical rod under gravity, friction and external pulse effect, leading to thermal expansion of the rod. The dynamics of the system under consideration corresponds to the behavior of a fuel element (FE) in a pulsed reactor and is related to the dynamic stability of the processes occurring in it.The FE dynamics is described by inhomogeneous linear differential equation of the fourth order with non-constant coefficients. Initial boundary conditions are not smooth since they correspond to the instant heating of the part of the FE surface exposed to neutron pulse radiation. The general solution of the homogeneous linear equation can be found using the concept of generalized functions expressed as a series expansion in terms of coordinate eigenfunctions dependent on p parameter. The p parameter is related with the FE mass and at some certain values results in bifurcation points of the boundary value problem for eigenfunctions. The partial solution can be found in several ways: using the Fourier transform, the method of Green's function, and in terms of series expansion by eigenfunctions.Eigenfunction expansion coefficients in an explicit form have been obtained and the numerical solution accuracy has been estimated for some important particular cases. The results of the obtained exact solutions appear to be very close to the results of the ANSYS numerical estimations. In the future the obtained exact solutions will be used as input data to simulate self-consistent system dynamics of more than 400 FE. Therefore, the advantage of the exact solution in practical use is that its calculation is much faster as compared with the finite element method done with ANSYS.

A new accurate solution method for steady forced convection in an unbounded flow around a circular cylinder in continuum and slip regimes

A new accurate numerical solution method for steady forced convection in an unbounded bidimensional flow around a circular cylinder is presented, addressing both the no-slip and slip regimes for Reynolds numbers and Knudsen numbers within the range 0.01 ≤ Re ≤ 47 and 0 ≤ Kn ≤ 0.1. The challenge of singular behaviour in infinite domains, which degrades numerical precision, is effectively mitigated by our new approach, unlike previous methods that truncate the flow domain. This method provides complete and correct description of the flow near the cylinder and in the far field perfectly reproducing the asymptotic solutions at large distances. We also present a new effective analytical solution for Oseen flow with slip effects, accurately representing flow at low Reynolds numbers. The validity of our numerical solution is confirmed by extensive comparisons with analytical solutions and numerical and experimental data from the literature. Additionally, we derive a new accurate correlation formula for the Nusselt number in slip flow regimes, involving both scenarios with and without temperature jump. This formula is crucial for engineering applications, especially in constant temperature hot-wire anemometry, as it avoids the requirement to correct heat transfer measurements due to slip and temperature jump effects when using very small diameter wires.

Adopted spectral tau approach for the time-fractional diffusion equation via seventh-kind Chebyshev polynomials

This study utilizes a spectral tau method to acquire an accurate numerical solution of the time-fractional diffusion equation. The central point of this approach is to use double basis functions in terms of certain Chebyshev polynomials, namely Chebyshev polynomials of the seventh-kind and their shifted ones. Some new formulas concerned with these polynomials are derived in this study. A rigorous error analysis of the proposed double expansion further corroborates our research. This analysis is based on establishing some inequalities regarding the selected basis functions. Several numerical examples validate the precision and effectiveness of the suggested method.