Numerical Solution Scheme Research Articles

Non-Standard Finite Difference Schemes for Solving Singular Lane-Emden Equation

In this paper we construct Non-Standard finite difference schemes (NSFD) for numerical solution of nonlinear Lane-Emden type equations which are nonlinear ordinary dierential equations on semi-infinite domain. They are categorized as singular initial value problems. This equation describes a variety of phenomena in theoretical physics and astrophysics. The presented schemes are obtained by using the Non-Standard finite difference method. The use of NSFD method and its approximations play an important role for the formation of stable numerical methods. The main advantage of the schemes is that the algorithm is very simple and very easy to implement. Thus, this method may be applied as a simple and accurate solver for ODEs and PDEs and it can also be utilized as an accurate algorithm to solve linear and nonlinear equations arising in physics and other fields of applied mathematics. Illustrative examples have been discussed to demonstrate validity and applicability of the technique and the results have been compared with the exact solutions.

Coupled Multiphysics Problems as Market Place for Competing Autonomous Software Agents

A novel approach is presented applying capabilities of software agent-based programming to numerical field computations. Distributed and autonomous calculation resources are used in analogy to a machine-to-machine system application for setting up and solving coupled simulations respecting the individual machine capabilities. Within the presented system, the involved machines interact cooperatively and comparatively. No central unit takes any influence to the solution process. Redundant resources do automatically and proactively add variance to the solution process by modifying their partial tasks based on negotiations between the involved resources. In that way, different numerical methods, solution strategies, and solver configurations are automatically evaluated as examples of the variety within coupled simulations. Partial results are dynamically integrated into an overall solver sequence, demonstrated on the coupled electric field computation of a patch antenna array. A quick provision of results is shown contemplating changes at calculation resource and vague user decisions for a nonlinear and bidirectionally coupled simulation of a 3-D multiphysics problem.

A numerical scheme for solutions of a class of nonlinear differential equations

In this paper, a collocation method based on Bessel functions of the first kind is presented to compute the approximate solutions of a class of high-order nonlinear differential equations under the initial and boundary conditions. First, the matrix forms of the Bessel functions of the first kind and their derivatives are constructed. Second, by using these matrix forms, collocation points and the matrix operations, a nonlinear differential equation problem is converted to a system of nonlinear algebraic equations. The solutions of this system give the coefficients of the assumed approximate solution. To demonstrate the validity and applicability of the technique, numerical examples are included and comparisons are made with existing results. The results show the efficiency and accuracy of the present work.

On the performance of four methods for the numerical solution of ecologically realistic size‐structured population models

Summary Size‐structured population models (SSPMs) are widely used in ecology to account for intraspecific variation in body size. Three characteristic features of size‐structured populations are the dependence of life histories on the entire size distribution, intrinsic population renewal through the birth of new individuals, and the potential accumulation of individuals with similar body sizes due to determinate or stunted growth. Because of these three features, numerical methods that work well for structurally similar transport equations may fail for SSPMs and other transport‐dominated models with high ecological realism, and thus their computational performance needs to be critically evaluated. Here, we compare the performance of four numerical solution schemes, the fixed‐mesh upwind (FMU) method, the moving‐mesh upwind (MMU) method, the characteristic method (CM), and the Escalator Boxcar Train (EBT) method, in numerically solving three reference problems that are representative of ecological systems in the animal and plant kingdoms. The MMU method is here applied for the first time to SSPMs, whereas the three other methods have been employed by other authors. Our results show that the EBT method performs best, except for one of the three reference problems, in which size‐asymmetric competition affects individual growth rates. For that reference problem, the FMU method performs best, closely followed by the MMU method. Surprisingly, the CM method does not perform well for any of the three reference problems. We conclude that life‐history features should be carefully considered when choosing the numerical method for analyzing ecologically realistic size‐structured population models.

Three-dimensional oil dispersion model in the Campos Basin, Brazil

ABSTRACTThis paper presents the physical and mathematical formulation of a three-dimensional oil dispersion model that calculates the trajectory from the seafloor to the sea surface, its assumptions and constraints. It was developed by researchers who are familiar with oil spill dispersion and mathematical analysis. Oil dispersion is calculated through two computational routines. The first calculates the vertical dispersion along the water column and resamples the droplets when the oil reaches the surface. The second calculates the surface displacement of the spill. This model is based on the Eulerian approach, and it uses numerical solution schemes in time and in space to solve the equation for advective–diffusive transport. A case study based on an actual accident that happened in the Campos Basin, in Rio de Janeiro state, considering the instant spill of 1000 m3 was used to evaluate the proposed model. After calculating the vertical transport, it was estimated that the area covered by the oil spill on the surface was about 35,685 m². After calculating the dispersion at the surface, the plume area was estimated as 20% of the initial area, resulting in a final area of 28,548 m².

Nonstandard finite difference schemes for numerical solution of the fractional neutron point kinetics equations

In this paper, our purpose is to find approximate solutions of fractional neutron point kinetic equations by using non-standard finite difference method. The fractional neutron point kinetic equations are modelled with average one group of delayed neutron precursors and the fractional derivative is given in the form of Grunwald-Letnikov. The efficiency and reliability of the suggested approach are proved by some numerical experiments for critical reactivity, supercritical reactivity and subcritical reactivity for various values of fractional order. It is found that the nonstandard finite difference method (NSFDM) is preferable than the standard finite difference method (SFDM). Also, the stability of the numerical scheme is investigated. The stability range of the step size is introduced for different values of the anomalous diffusion order (α) and of the relaxation time (τ). Numerical results and graphs for neutron flux for different values of the anomalous order and of the relaxation time are shown and compared with the classical solutions.

A continuity-preserving and divergence-cleaning algorithm based on purely and damped hyperbolic Maxwell equations in inhomogeneous media

In the numerical solution of Maxwell's equations for dynamic electromagnetic fields, the two Gauss's laws are usually not considered since they are a natural consequence of Faraday's and Maxwell–Ampère's laws if the charge conservation law is satisfied. However, when the charge conservation law is not satisfied, the numerical errors of Gauss's laws will increase unbounded, leading to numerical instability or breakdown. Unfortunately, the charge conservation law can be easily violated in self-consistent wave–particle simulations. In the meantime, the violation of Gauss's laws will also result in an increased error in the normal flux continuity. In the simulations of pure electromagnetic problems, the satisfaction of tangential field continuity across a material interface is sufficient to yield accurate numerical results. However, in a self-consistent wave–particle simulation, the normal components are as important as the tangential components, since they are critical in predicting the particle kinetics. In this paper, a divergence-cleaning method is presented to enforce Gauss's laws and normal flux continuity by introducing auxiliary variables and damping terms into Maxwell's equations in inhomogeneous media, which yield hyperbolic and mixed hyperbolic–parabolic Maxwell equations. The numerical solution schemes of the resulting purely and damped hyperbolic Maxwell equations are introduced under the framework of the discontinuous Galerkin time-domain method. It is shown through several numerical examples that the proposed method is able to preserve the continuity conditions for both tangential field components and normal flux components, and is effective in cleaning the numerical errors in both Gauss's laws.

A sequential approach for numerical simulation of two-phase multicomponent flow with reactive transport in porous media

We develop a new scheme for numerical solution of immiscible compressible two-phase flow in porous media with geochemistry. The problem is modelled by the mass balance law for each phase, Darcy–Muskat’s law, and the capillary pressure law. Coupling with chemistry occurs through reactions rates. These rates can be either given nonlinear functions of concentrations in the case of kinetic chemical reactions or unknown for equilibrium chemical reactions. Each kinetic reaction produces an ordinary differential equation while each equilibrium reaction gives rise to a mass action law that is an algebraic relation that links the activities of concerned species. An implicit finite volume scheme is applied to solve the two-phase flow equations, which is then sequentially coupled to a method for solving the reactive transport problem. More precisely, we used firstly the module 2p2c implemented in the parallel open-source simulator DuMuX to solve a simplified two-phase two-component flow with two dominant species without chemistry. Secondly, we have developed and integrated a reactive transport module in the DuMuX framework to deal with the other species using a sequential iterative approach (SIA) where transport, equilibrium chemical reactions and kinetic chemical reactions are solved sequentially. A new module for transport and a code using the GSL library for the chemical problem have been coupled. Finally, our approach has been validated by solving several test cases. Here we will present two benchmark tests to demonstrate the ability of our method to approximate solutions of single and two-phase flows with reactive transport in heterogeneous porous media.

A family of positivity preserving schemes for numerical solution of Black–Scholes equation

When one solves the Black–Scholes partial differential equation, it is of great important that numerical scheme to be free of spurious oscillations and satisfy the positivity requirement. With positivity, we mean, the component non-negativity of the initial vector, is preserved in time for the exact solution. Numerically, such property for fully implicit scheme is not always satisfied by approximated solutions and they generate spurious oscillations in the presence of discontinuous payoff. In this paper, by using the nonstandard discretization strategy, we propose a new scheme that is free of spurious oscillations and satisfies the positivity requirement.

Non-Newtonian versus numerical rheology: Practical impact of shear-thinning on the prediction of stable and unstable flows in intracranial aneurysms.

Computational fluid dynamics (CFD) shows promise for informing treatment planning and rupture risk assessment for intracranial aneurysms. Much attention has been paid to the impact on predicted hemodynamics of various modelling assumptions and uncertainties, including the need for modelling the non-Newtonian, shear-thinning rheology of blood, with equivocal results. Our study clarifies this issue by contextualizing the impact of rheology model against the recently demonstrated impact of CFD solution strategy on the prediction of aneurysm flow instabilities. Three aneurysm cases were considered, spanning a range of stable to unstable flows. Simulations were performed using a high-resolution/accuracy solution strategy with Newtonian and modified-Cross rheology models and compared against results from a so-called normal-resolution strategy. Time-averaged and instantaneous wall shear stress (WSS) distributions, as well as frequency content of flow instabilities and dome-averaged WSS metrics, were minimally affected by the rheology model, whereas numerical solution strategy had a demonstrably more marked impact when the rheology model was fixed. We show that point-wise normalization of non-Newtonian by Newtonian WSS values tended to artificially amplify small differences in WSS of questionable physiological relevance in already-low WSS regions, which might help to explain the disparity of opinions in the aneurysm CFD literature regarding the impact of non-Newtonian rheology. Toward the goal of more patient-specific aneurysm CFD, we conclude that attention seems better spent on solution strategy and other likely "first-order" effects (eg, lumen segmentation and choice of flow rates), as opposed to "second-order" effects such as rheology.

Numerical model of bioaerosol transformation in the atmosphere

A nonstationary mathematical model of bioaerosol dynamics is considered. It is based on nonlinear integral-differential equations that describe coagulation, condensation, and evaporation processes versus particle sizes. A definitely positive numerical scheme for solution of the problem of aerosol transformation in the atmosphere is presented. The model is numerically compared with the models that describe individual processes in the composition of the former. The relative contribution of each process in the overall dynamics of aerosol populations is studied in numerical experiments.

Penny-shaped crack in elastic medium with surface energy effects

In this paper, a penny-shaped crack in an infinite elastic medium subjected to vertical pressure loading at the crack surface under the influence of surface stress is considered. The Gurtin–Murdoch continuum theory of elastic material surfaces is adopted, and the Hankel integral transform is employed to solve this axisymmetric boundary value problem. A set of simultaneous dual integral equations is solved by employing an appropriate numerical solution scheme. Selected numerical results are presented to portray the influence of the surface stress on the elastic field. Numerical results reveal that the surface stress has a significant influence on both stress and displacement fields. It is also found that the material becomes tougher with the presence of surface stress. In addition, the elastic field also shows size-dependent behavior deviating from the classical crack solution. The solutions presented in this study provide fundamental understanding of the influence of surface stress on fracture mechanics problems. It can also be used as a benchmark solution for the development of numerical techniques, such as FEM and BEM, for analysis of more complicated problems associated with cracks under the influence of surface stress.

An operational Haar wavelet collocation method for solving singularly perturbed boundary-value problems

The basic aim of this study is to introduce and describe a numerical scheme for the approximate solutions of the one-dimensional singularly perturbed boundary-value problems. The method is based on Haar wavelets and its main characteristic is that, it converts the given problem into a system of algebraic equations that can be solved easily with any of the usual methods. Another distinguishing feature of the this method is that unlike several other numerical methods, it does not require conversion of a boundary value problem into initial-value problem and hence eliminates the possibility of unstable solutions. To show the accuracy and the efficiency of the method, several benchmark problems are implemented and the comparisons are given with other methods existing in the recent literature. The results of numerical tests confirm that the Haar wavelet collocation method is superior to other existing ones and is highly accurate.

Isotropic Total Variation Regularization of Displacements in Parametric Image Registration.

Spatial regularization is essential in image registration, which is an ill-posed problem. Regularization can help to avoid both physically implausible displacement fields and local minima during optimization. Tikhonov regularization (squared l2 -norm) is unable to correctly represent non-smooth displacement fields, that can, for example, occur at sliding interfaces in the thorax and abdomen in image time-series during respiration. In this paper, isotropic Total Variation (TV) regularization is used to enable accurate registration near such interfaces. We further develop the TV-regularization for parametric displacement fields and provide an efficient numerical solution scheme using the Alternating Directions Method of Multipliers (ADMM). The proposed method was successfully applied to four clinical databases which capture breathing motion, including CT lung and MR liver images. It provided accurate registration results for the whole volume. A key strength of our proposed method is that it does not depend on organ masks that are conventionally required by many algorithms to avoid errors at sliding interfaces. Furthermore, our method is robust to parameter selection, allowing the use of the same parameters for all tested databases. The average target registration error (TRE) of our method is superior (10% to 40%) to other techniques in the literature. It provides precise motion quantification and sliding detection with sub-pixel accuracy on the publicly available breathing motion databases (mean TREs of 0.95 mm for DIR 4D CT, 0.96 mm for DIR COPDgene, 0.91 mm for POPI databases).

Technology of Gasification of Liquefied Natural Gas

A flow diagram of gasification of a cryogenic liquid, which is based on the utilization of the liquid′s internal energy to obtain a vapor phase, has been presented. The limiting steam fractions of the two-phase flow in a gasifier have been evaluated as applied to the problems of gasification of methane. Consideration has been given to the conditions of phase separation in the field of mass forces. A numerical scheme of solution of a system of gasdynamic equations for the two-phase flow in a cylindrical coordinate system in a three-dimensional formulation has been implemented. The results of numerical modeling of the conditions of precipitation of particles with a diameter of 2 to 10 μm from a swirling dispersed flow have been presented; the role of the particle size in the dynamics of the process of phase separation has been established.

Lagrangian approach to modeling unsteady gas-liquid flow in a well

The purpose of this paper is to develop a numerical method of solving the problem of evolution of the finite gas volume that entered in a liquid flow at a set flow rate. The drift- flux model is used as gas-liquid mixture equations. The velocities of both phases, mixture and gas, are related by the Zuber-Findlay equation which coefficients depend on flow regime and gas void fraction. Lagrangian coordinates are used to simplify the initial equations. The numerical solution scheme is proposed. The numerical solution of the Riemann problem is verified by comparison with the exact self-similar solution. The model and numerical method efficiency is illustrated by examples of gas kick calculations in a vertical well.

Bilayer-thickness-mediated interactions between integral membrane proteins.

Hydrophobic thickness mismatch between integral membrane proteins and the surrounding lipid bilayer can produce lipid bilayer thickness deformations. Experiment and theory have shown that protein-induced lipid bilayer thickness deformations can yield energetically favorable bilayer-mediated interactions between integral membrane proteins, and large-scale organization of integral membrane proteins into protein clusters in cell membranes. Within the continuum elasticity theory of membranes, the energy cost of protein-induced bilayer thickness deformations can be captured by considering compression and expansion of the bilayer hydrophobic core, membrane tension, and bilayer bending, resulting in biharmonic equilibrium equations describing the shape of lipid bilayers for a given set of bilayer-protein boundary conditions. Here we develop a combined analytic and numerical methodology for the solution of the equilibrium elastic equations associated with protein-induced lipid bilayer deformations. Our methodology allows accurate prediction of thickness-mediated protein interactions for arbitrary protein symmetries at arbitrary protein separations and relative orientations. We provide exact analytic solutions for cylindrical integral membrane proteins with constant and varying hydrophobic thickness, and develop perturbative analytic solutions for noncylindrical protein shapes. We complement these analytic solutions, and assess their accuracy, by developing both finite element and finite difference numerical solution schemes. We provide error estimates of our numerical solution schemes and systematically assess their convergence properties. Taken together, the work presented here puts into place an analytic and numerical framework which allows calculation of bilayer-mediated elastic interactions between integral membrane proteins for the complicated protein shapes suggested by structural biology and at the small protein separations most relevant for the crowded membrane environments provided by living cells.

Reliability-based performance optimization of TMD for vibration control of structures with uncertainty in parameters and excitation

Recent development of system identification using Bayesian models or stochastic filtering provides probabilistic descriptions (i.e., probability density function or statistical parameters like mean and variance) of the identified model parameters (e.g., mass, stiffness, and damping). Optimal design of passive controllers for these systems whose parameters are uncertain has remained an open problem. With this in view, the present study aims to develop numerical solution scheme for the optimal design of tuned mass damper (TMD) operating in uncertain environment. Deterministic design of TMD in these cases suffers detuning as the system parameters are random. Thus, a reliability-based design optimization (RBDO) scheme is presented in this paper for better performance of the TMD when exposed to uncertainties. To solve the RBDO problem, response surface methodology is used along with the moving least squares technique. Dual response surfaces are used for separate handling of optimization and reliability analysis. First response surface performs optimization of the design variables of TMD, while the second response surfaces are used for the estimation of the statistical properties like mean and variance to satisfy the constrained conditions. Numerical analysis is presented to show the effectiveness of the proposed algorithm for RBDO of single degree of freedom-TMD system as a proof of concept. The proposed meta-model-based algorithm can be applied for the optimal design of controller for large structures where conventional technique may face difficulty to handle both optimization and uncertainty quantification simultaneously. Copyright © 2016 John Wiley & Sons, Ltd.

Thermo-electro-mechanical vibration of postbuckled piezoelectric Timoshenko nanobeams based on the nonlocal elasticity theory

In this study, free vibration behavior of piezoelectric Timoshenko nanobeams in the vicinity of postbuckling domain is investigated based on the nonlocal elasticity theory. It is assumed that the piezoelectric nanobeam is subjected to an axial compression force, an applied voltage and a uniform temperature change. Using Hamilton principle, the governing differential equations of motion incorporating von Kármán geometric nonlinearity and the corresponding boundary conditions are derived and then discretized on the basis of generalized differential quadrature (GDQ) scheme. After solving the parameterized equations using Newton–Raphson technique, a dynamic analysis based on a numerical solution strategy is performed to predict the natural frequencies of piezoelectric nanobeams associated with both prebuckling and postbuckling domains. Numerical results are presented to study the effects of nonlocal parameter, temperature rise and external electric voltage on the size-dependent vibration behavior of piezoelectric nanobeams with clamped–clamped (C–C), clamped-simply supported (C-SS) and simply supported-simply supported (SS-SS) end conditions. It is demonstrated that these parameters may shift the postbuckling domain to higher or lower applied axial loads.

Quantum impurities in channel mixing baths

We propose a versatile strategy for numerical renormalization group solution of general channel-mixing Kondo and Anderson models beyond previous reach, opening the door toward broad applications in protocol non-perturbative machineries, such as dynamical cluster approximation and cluster dynamical mean field theory, for strongly correlated electron systems. We illustrate the strategy by investigating the quantum phase transitions in two quantum impurity models with cases untouched before.