Spectral Collocation Method Research Articles

Flow of a limited stress fluid between plates rotating about different axes

In this paper we study the flow of a particular type of non-Newtonian fluids generated by the rotation of parallel infinite plates about distinct axes. The constitutive law of these fluids mimics the response of a class of seemingly viscoplastic or “yield stress” materials in which the norm of the stress is bounded by a critical threshold (limited stress fluids). We assume that the plates rotate with the same angular velocity and we show that, in this case, the mathematical problem can be reduced to solving a BVP where the unknowns are the coordinates of the center of rotation. We solve the problem numerically (by means of a spectral collocation method), and we investigate the dependence of the locus of the center of rotation on the material parameters. We prove that, even for small Reynolds numbers, the flow may exhibit boundary layers depending on the particular choice of the parameters.

Numerical treatment via the spectral collocation method for Casson–Williamson nanofluid flow due to a stretching sheet with slip conditions

The main goal of the research is to investigate the effects of temperature and concentration slip on the MHD Casson–Williamson nanofluid flow through a porous sheet, with a focus on viscous dissipation. This research has a novel approach, as it seeks to incorporate various factors that have not been previously studied in this context. Moreover, an inclined magnetic field from the outside is applied to the stretched surface. Besides this, it is considered that the viscosity of nanofluids depends on temperature while all other physical characteristics are taken to be constant. It is hypothesized that the expanding surface that stretched exponentially is what caused the flow motions of the nanofluid. There are several engineering applications for this type of study, which is based on MHD non-Newtonian nanofluid flow across a stretched surface with heat generation and viscous dissipation. They include solar energy storage, energy distribution, chemical reactors, and the manufacturing of polymers. To examine the heat transmission and mass transfer rates, tools like thermophoresis and Brownian motion were implemented. The flow problem is formally represented as a set of nonlinear PDEs that are then, through the use of dimensionless variables, converted into a set of ODEs. To numerically obtain the solution to the problem, the shifted Chebyshev polynomials of the third-kind approximation along with the spectral collocation technique are utilized. This process transforms the existing model into an algebraic equation system that was created as a restricted optimization problem, which is then optimized to obtain the solution and the unknown coefficients. For a variety of pertinent parameter values, the graphic and tabular representations of the concentration, temperature, velocities, rate of heat mass transfer, and shear stresses of nano-fluids at the surface of the sheet are shown. Finally, there is a remarkable amount of agreement between the earlier investigation and the comparison research that was conducted. The findings suggest that higher values of the magnetic parameter, Casson parameter, viscosity parameter, and suction parameter lead to a decrease in both velocity and boundary layer thickness. Furthermore, an increase in the viscosity parameter, Casson parameter, and magnetic parameter results in elevated values for both temperature and concentration.

Legendre-tau Chebyshev collocation spectral element method for Maxwell's equations with material interfaces of two dimensional transverse magnetic mode

Legendre-tau Chebyshev collocation spectral element method for Maxwell's equations with material interfaces of two dimensional transverse magnetic mode

Spectral collocation methods for fractional multipantograph delay differential equations*

Spectral collocation methods for fractional multipantograph delay differential equations*

Convective heating and mass transfer in Buongiorno model of nanofluid using spectral collocation method of shifted Chebyshev polynomial

In this article, the investigation is made to shed a light on the influence of convective boundary conditions on the boundary layer flow over a linearly stretching flat surface. The equations (partial differential equations) describing the model are transformed into system of nonlinear ordinary differential equations using similarity transformations. Equations contain various non-dimensional flow characterizing numbers viz. Prandtl number Pr, Lewis number Le, Biot number Bi, Brownian motion parameter Nb and thermophoresis parameter Nt. The influence of these parameters on thermal boundary layer, concentration distribution and temperature are analyzed in detail, by solving the equations using novel Shifted Chebyshev collocation method. The computed results, reduced Nusselt number, reduced Sherwood number, surface temperature and concentration profiles as functions of dimensionless numbers are validated by comparing the predicted results with available earlier findings (using other methods). To assert the convergence and stability of the scheme used (for much larger, but moderate, parameters values), predicted results are presented in various tabular forms. For presenting finer details of the computed values some results are also given graphically. The innovative semi-numerical scheme is robust and efficient compared with other conventional methods, used in previous studies and enables the analysis of the complex problem adequately.

The threshold of instability of Taylor–Couette flow of a Newtonian fluid in quasi-periodic modulation with two immense frequencies using spectral Chebychev collocation methods

This study investigates the dynamic flow of a Newtonian fluid through two coaxial cylinders, each rotating at speeds Ω1(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\Omega }_{1}(t)$$\\end{document} (inner cylinder) and Ω2(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\Omega }_{2}(t)$$\\end{document} (outer cylinder). We derive equations of motion for disturbances in balance, yielding a controlled system characterized by parameters such as Taylor number, wave number, frequency ratio, and interior cylinder frequency. We introduce numerical techniques for solving this system, employing spectral Chebychev collocation for spatial resolution and a combined approach of Floquet theory and Runge–Kutta method for temporal resolution. Our refined approach enables comprehensive analysis of fluid dynamics within the rotating coaxial cylinders, showcasing the interplay of various control parameters.

A hybrid technique based on Lucas polynomials for solving fractional diffusion partial differential equation

This paper presents a new numerical technique to approximate solutions of diffusion partial differential equations with Caputo fractional derivatives. We use a spectral collocation method based on Lucas polynomials for time fractional derivatives and a finite difference scheme in space. Stability and error analyses of the proposed technique are established. To demonstrate the reliability and efficiency of our new technique, we applied the method to a number of examples. The new technique is simply applicable, and the results show high efficiency in calculation and approximation precision.

A spatial local method for solving 2D and 3D advection-diffusion equations

PurposeThis study aims to derive a novel spatial numerical method based on multidimensional local Taylor series representations for solving high-order advection-diffusion (AD) equations.Design/methodology/approachThe parabolic AD equations are reduced to the nonhomogeneous elliptic system of partial differential equations by utilizing the Chebyshev spectral collocation method (ChSCM) in the temporal variable. The implicit-explicit local differential transform method (IELDTM) is constructed over two- and three-dimensional meshes using continuity equations of the neighbor representations with either explicit or implicit forms in related directions. The IELDTM yields an overdetermined or underdetermined system of algebraic equations solved in the least square sense.FindingsThe IELDTM has proven to have excellent convergence properties by experimentally illustrating both h-refinement and p-refinement outcomes. A distinctive feature of the IELDTM over the existing numerical techniques is optimizing the local spatial degrees of freedom. It has been proven that the IELDTM provides more accurate results with far fewer degrees of freedom than the finite difference, finite element and spectral methods.Originality/valueThis study shows the derivation, applicability and performance of the IELDTM for solving 2D and 3D advection-diffusion equations. It has been demonstrated that the IELDTM can be a competitive numerical method for addressing high-space dimensional-parabolic partial differential equations (PDEs) arising in various fields of science and engineering. The novel ChSCM-IELDTM hybridization has been proven to have distinct advantages, such as continuous utilization of time integration and optimized formulation of spatial approximations. Furthermore, the novel ChSCM-IELDTM hybridization can be adapted to address various other types of PDEs by modifying the theoretical derivation accordingly.

Direct and inverse problems for a 2D heat equation with a Dirichlet–Neumann–Wentzell boundary condition

In this paper we present the inverse problem of determining a time dependent heat source in a two-dimensional heat equation accompanied with Dirichlet–Neumann–Wentzell boundary conditions. The model is of significant practical importance in applications where the time dependent internal source is to be controlled from total energy measurements in the case when the boundary is defined with partially absorbing (Dirichlet), partially reflecting (Neumann), and partially having the capacity for storing heat (Wentzell) conditions. The considered spatial domain is a rectangle which allows the application of a Fourier series analysis to the solution of the problem whose auxiliary spectral problem is a novel eigenproblem. This eigenvalue problem is defined using the two-dimensional Laplace operator where the spectral parameter resides in one of the accompanying boundary conditions. Under this setting we study the well-posedness of both the direct and inverse problems. In addition, we use the results obtained on the well-posedness to establish the numerical solvability of all the problems considered in the paper. In order to achieve this, we make use of a Chebyshev spectral collocation method for the spatial discretisation and a backward differentiation formula for the temporal discretisation. All the numerical results are shown to be consistent with the theory and in good agreement with the analytical solutions.

Electrokinetic Thin-Film Model for Electrowetting: The Role of Bulk Charges.

The electrowetting behavior of a charge-carrying sessile droplet is relevant to applications such as point-of-care diagnostics. Often biomedical assays involve droplets that contain charged molecules such as dissolved ions, proteins, and DNA. In this work, we develop a reduced-order electrokinetic model for electrowetting of such a charge-carrying droplet under a parallel-plate electrode configuration. An inertial-lubrication model based on the weighted residual integral boundary layer (WRIBL) technique is used to obtain evolution equations that describe the spatiotemporal evolution of the fluid-air interface and the depth-integrated flow rate. The solutions to the evolution equations are obtained numerically by using the spectral collocation method. We investigate the role of domain and surface charges, characterized by the Debye length, on droplet wetting. Under low relaxation timescales, both droplet deformation and wetting alteration under an AC field are shown to be equivalent to that under a root-mean-square (RMS) DC field. We show that an electrolytic sessile droplet can exhibit a larger deformation in comparison to the two asymptotic limits of a perfect conductor and a perfect dielectric droplet, corresponding, respectively, to very low and high Debye lengths. The effects of several other parameters such as the inherent equilibrium wettability, permittivity ratio, and electric field strength are also investigated.

Non-Newtonian nanofluid flow across an exponentially stretching sheet with viscous dissipation: numerical study using an SCM based on Appell–Changhee polynomials

The objective of this article is to investigate how the properties of a non-Newtonian Williamson nanofluid flow, which occurs due to an exponential stretching sheet placed in a porous medium, are influenced by heat generation, viscous dissipation, and magnetic field. This study focuses on analyzing the heat transfer process by considering the impact of temperature on the thermal conductivity and viscosity of Williamson nanofluids. Additionally, the research significantly contributes by investigating the flow characteristics of these nanofluids when influenced by slip velocity. Using the spectral collocation method (SCM), the equations that describe the current problem are transformed into a collection of ordinary differential equations and then solved. The SCM proposed here basically depends on the properties of the Appell-type Changhee polynomials (ACPs). First, with the aid of ACPs, we give an approximate formula of the derivatives for the approximated functions. Through this procedure, the provided model is transformed into a nonlinear set of algebraic equations. Physical factors of interest, such as skin friction, the Nusselt number, and the Sherwood number, are explained using tabular expressions. Data are displayed as graphs for the nanofluid’s velocity, temperature, and concentration. The primary findings showed that increasing the Williamson, magnetic, thermal conductivity, and Brownian parameters significantly improves the thermal field. Finally, testing the suggested method with specific cases from some past literature-based publications reveal a good degree of agreement.

Spectral and Haar wavelet collocation method for the solution of heat generation and viscous dissipation in micro-polar nanofluid for MHD stagnation point flow

The aim and significance of paper presents, the semi-numerical investigation of magnetohydrodynamic flow of micropolar nanofluid with stagnation point is carried out under the influence of viscous dissipation and heat generation. The micropolar nanofluids are electrically conducting non-Newtonian fluids. The important applications of these fluids are observed in many research areas viz. bioengineering, biofuels and biomedical sectors etc. The appropriate similarity transformations are used to transform the governing equations into system of coupled nonlinear ordinary differential equations and are solved by using shifted Chebyshev collocation method and Haar wavelet collocation method. The variations in velocity, angular velocity, temperature and concentration profiles under the impact of various physical parameters, characterizing the flow field are discussed and are presented via graphs and tables. Temperature enhancement occurs with increment in each parameter except for Prandtl number. The concentration near the surface decreases with increment in the values of parameters and gradually it increases, except for Prandtl number and Schmidt number. The reverse trend of heat transfer occurs​ near a surface, when the dominance of stream velocity over stretching velocity is observed.

A fast time-stepping method based on the [formula omitted]-version spectral collocation method for the nonlinear fractional delay differential equation

In this paper, we propose a fast time-stepping method based on the hp-version spectral collocation method for the nonlinear fractional delay differential equation. To reduce the interaction between the subintervals caused by the delay term, a mesh of bidirectional partition is designed, such that each subinterval after delay falls exactly into a single preceding subinterval. Moreover, with the aid of the sum-of-exponentials approximation for the kernel function (t−s)α−1, a fast time-stepping method is developed to deal with the weakly singular integral, which overcomes the weak singularity of the integral kernel, and greatly reduces the cost for the computation of the history integral. In addition, the hp-convergence of the suggested method is fully analyzed and characterized, which implies that the interplay between h and p can significantly enhance the numerical accuracy. Numerical experiments confirm the theoretical expectations.

Chebyshev collocation method for solving second order ODEs using integration matrices

The spectral collocation method for solving two-point boundary value problems for second order differential equations is implemented, based on representing the solution as an expansion in Chebyshev polynomials. The approach allows a stable calculation of both the spectral representation of the solution and its pointwise representation on any required grid in the definition domain of the equation and additional conditions of the multipoint problem. For the effective construction of SLAE, the solution of which gives the desired coefficients, the Chebyshev matrices of spectral integration are actively used. The proposed algorithms have a high accuracy for moderate-dimension systems of linear algebraic equations. The matrix of the system remains well-conditioned and, with an increase in the number of collocation points, allows finding solutions with ever-increasing accuracy.

On the numerical evaluation for studying Ohmic dissipation and thermal conductivity impacts on the flow of Casson fluid

Analysis of a steady flow of a viscous Casson fluid subject to Ohmic dissipation and an induced magnetic field is the main goal here. Through a stretched vertical sheet, the flow is managed. The energy equation is explained in a thermodynamical system along with Ohmic heating. By making the assumption that the viscosity and thermal conductivity are temperature-dependent as described here, an accurate assessment of the heat transfer rate may be carried out. Our research also addresses the effect of slip velocity. There are many applications for thermal flow fields subject to limited spaces and heated walls, including electronic cooling systems, furnaces, cooling towers, and thermal individualities in buildings and rooms, to highlight a few. The governing equations of the current model are generated mathematically in the form of partial differential equations by applying the boundary layer approximations along with these assumptions. Through appropriate transformations, a nonlinear system of differential equations is produced. The approximate solution is then obtained by using the Chebyshev spectral collocation method. The proposed problem is reduced to a nonlinear system of algebraic equations with the help of the properties of Chebyshev polynomials of the third-kind. Diagrams are therefore used to illustrate the significant impact of all controlling parameters on the fluid flow. Further, the local Nusselt number and the skin-friction coefficients’ numerical values are also computed and examined. Results show that the Casson fluid motion is strengthened by the electric and mixed convection characteristics, whereas the magnetic, the slip velocity and viscosity parameters impede the flow motion. Also, the distribution of temperature is amplified by a larger Eckert number.

Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations

We prove exponential expression rates of deep operator networks (deep ONets) between infinite-dimensional spaces that emulate the coefficient-to-solution map of linear, elliptic, second-order, divergence-form partial differential equations (PDEs). In particular, we consider problems set in -dimensional periodic domains, , with analytic right-hand sides and coefficients. Our analysis covers diffusion-reaction problems, parametric diffusion equations, and certain elliptic systems such as linear isotropic elastostatics in heterogeneous materials. In the constructive proofs, we leverage the exponential convergence of spectral collocation methods for boundary value problems whose solutions are analytic. In the present periodic and analytic setting, this follows from classical elliptic regularity. Within the branch and trunk ONet architecture of Chen and Chen [IEEE. Trans. Nural Netw., 4 (1993), pp. 910–918] and of Lu et al. [Nat. Mach. Intell., 3 (2021), pp. 218–229], we construct deep ONets which emulate the coefficient-to-solution map to a desired accuracy in the norm, uniformly over the PDE coefficient set. The ONets have size , where is the approximation accuracy, for some depending on the physical space dimension.

Enhancing heat transfer in solar-powered ships: a study on hybrid nanofluids with carbon nanotubes and their application in parabolic trough solar collectors with electromagnetic controls

The aim of this research is to explore the use of solar-powered ships (SPS) as a means to reduce greenhouse gas emissions and fossil fuel dependency in the maritime industry. The study focuses on improving the heat transfer efficiency in SPS by employing hybrid nanofluids (HNF) containing carbon nanotubes (CNTs). Additionally, a novel approach utilizing renewable energy and electromagnetic control is proposed to enhance the performance of SPS. The research implements the non-Newtonian Maxwell type and Cattaneo–Christov heat flux model in parabolic trough solar collectors used for ships. The study conducts theoretical experiments and simulations to evaluate the thermal conductivity and viscosity of the CNT-based HNF. Various properties, including solar thermal radiation, viscous dissipation, slippery velocity, and porous media, are assessed to determine the effectiveness of thermal transport in SPS. The research employs similarity variables to simplify the complex partial differential equations into ordinary differential equations and solves them using the Chebyshev collocation spectral method. The results indicate that the MWCNT-SWCNT/EO hybrid nanofluid significantly improves the thermal conductivity, thereby enhancing heat transfer. The HNF exhibits an efficiency rate of approximately 1.78% with a minimum efficiency rate of 2.26%.

Numerical Simulation by Using the Spectral Collocation Method for Williamson Nanofluid Flow Over an Exponentially Stretching Sheet with Slip Velocity

The current research examines the rate of heat and mass transfer in MHD non-Newtonian Williamson nanofluid flow across an exponentially permeable stretched surface sensitive to heat generation/absorption and mass suction. The influences of Brownian motion and thermophoresis are included. In addition, the stretched surface is subjected to an angled outside magnetic field. This study incorporates the variable viscosity, viscous dissipation, and slip velocity. The fundamental rules of motion and heat transmission have been constructed mathematically to fit the current flow problem. By using appropriate self-similarity transformations, the supplied system of PDEs is transformed into a nonlinear system of ODEs. Here, we use the spectral collocation method with the help of Vieta-Lucas polynomials approximation. This procedure converts the present model to a system of algebraic equations which is developed as a constrained optimization problem, which is then optimized to get the solution and the unknown coefficients. Calculations are made for the skin friction, wall temperature gradient, and wall concentration gradient. By comparing our findings in some special cases to those in the literature, a review of the literature confirms the results described here.

Double-Diffusive Nonlinear Buoyancy Force Significance on Free Convective Chemically Reacting Fluids Flow Past Vertical Riga Surface

In this research, double-diffusive nonlinear buoyancy force effects on free convective chemically reacting fluids flow past a vertical plate in a Riga surface have been addressed numerically. Cattaneo-Christov theories, viscous dissipation, thermal radiation, thermophoresis and the Soret-Dufour mechanism were modelled in this paper. Also, thermal radiative heat flux was considered based on the Rosseland approximation because the fluid is optically thin. The flow equations were represented using partial differential equations (PDEs). These sets of PDEs were transformed into ordinary differential equations by utilizing suitable similarity variables. The transformed flow equations were numerically solved by employing the spectral collocation method. The outcomes show that enhancement in the velocity and temperature are observed due to an increase in Eckert number, thermal radiation and heat generation parameters. An increase in the relaxation heat and mass flux is noticed to have effects on the concentration and temperature gradient due to the presence of Soret-Dufour mechanisms. The present outcomes were compared with previously published works and are found to be in good agreement.

Mixed convection instability in a viscosity stratified flow in a vertical channel

The present study examines the linear instability characteristics of double-diffusive mixed convective flow in a vertical channel with viscosity stratification. The viscosity of the fluid is modeled as an exponential function of temperature and concentration, with an activation energy parameter determining its sensitivity to temperature variation. Three scenarios are considered: buoyancy force due to thermal diffusion only, buoyancy force due to temperature and solute acting in the same direction, and buoyancy force due to temperature and solute acting in opposite directions. A generalized eigenvalue problem is derived and solved numerically for the linear stability analysis via the Chebyshev spectral collocation method. The results indicate that higher values of the activation energy parameter lead to an increased flow stability. Additionally, when both buoyant forces act in opposite directions, the Schmidt number has both stabilizing and destabilizing effects across the range of activation energy parameters, similar to the case of pure thermal diffusion. Furthermore, the solutal-buoyancy-opposed base flow is found to be the most stable, while the solutal-buoyancy-assisted base flow is the least stable. As expected, an increase in the Reynolds number is shown to decrease the critical Rayleigh number.