Chebyshev Scheme Research Articles

Heat radiation absorption and irreversibility of electromagnetic Williamson hybridized Al2O3–CoFe2O4/H2O nanofluid: A concentrated power generation

The quest for alternative and ecosystem-friendly energy sources to reduce reliance on fossil fuels and minimize cost has stimulated studies on various energy-generating techniques. Solar energy is one of the renewable and low-cost power generation sources for domestic and industrial usage. To optimize the performance of solar power, nanofluids can be considered as the based material. This study focuses on the concentrated power generation through hybridization of electromagnetic Williamson aluminium oxide (Al2O3) and cobalt ferrite (CoFe2O4) nanofluids in water (H2O) with nonlinear radiation. Without the deformation of the material, the fluid is driven by nonlinear thermal convection and pressure gradient in a bounded device. The developed, transformed boundary value mathematical model is solved by the Chebyshev scheme coupled with the collocation integrating method. A theoretical analysis is carried out for the dependent-parameter sensitivities on the flow rate, heat distribution and entropy generation to enhance and maintain continuous power generation. The investigation revealed that the hybridized nanofluid enhanced the concentrated thermal power for about 12.23 %, and the fluid velocity field is raised by 3.45 % with increasing dependent parameters. The Williamson material term optimized the thermodynamic equilibrium of entropy generation.

A numerical study of tiny particles experiencing thermal and reaction migration in a micropolar fluid over a stretching sheet with Newtonian heating

ABSTRACT Substantial research on a micropolar fluid is currently being done. These novel materials offer various applications in bionic engineering and prosthetic design. The influences of cross-diffusive (diffusion-thermo and thermo-diffusion) on the stream of micropolar fluid via permeable stretching sheet under Newtonian heating are investigated in this study. The assumed mathematical model is then transformed using appropriate similarity transformations into its non-dimensional system. The resulting ordinary derivative formulation is solved through the spectral Chebyshev scheme with collocating integration. Graphically, the impacts of all critical factors on dimensionless velocity, microrotation, thermal distribution, and mass transfer are investigated. Tables are created to depict the interaction of various factors on engineering quantities. The outputs are compared with the accessible literature to demonstrate the effectiveness of the spectral Chebyshev schema, and excellent agreement is obtained.

An explicit iterative scheme for 3D multicomponent heat conducting flow simulation

This paper presents the development and testing of time integration scheme for the system of hydrodynamic equations of gas mixture. These equations take into account the phenomena of multicomponent diffusion and heat transfer. The governing equation system is discretized by the finite volume approach on three-dimensional unstructured grids. According to the algorithm, computation of any single time step is split into the sequence of hyperbolic and parabolic stages. The hyperbolic subtask is solved using the Godunov-type scheme. The parabolic subtask is solved by the explicit iterative Chebyshev scheme, which is algorithmically simple and does not involve tuning parameters. This stage addresses dissipative fluxes (viscosity, multicomponent diffusion and thermal conductivity). The number of the explicit iterations is determined by the convective time step and by the upper bound for the discrete diffusion operator. The resulting scheme ensures the fulfillment of the conservation laws at the discrete level. The computer code can be used in highly parallel computing for large-scale simulation. The proposed approach is recommended for applied problems in which convective and dissipative processes are in close interaction, particularly for problems of plasma physics and astrophysics.

PRICING AMERICAN OPTIONS WITH THE RUNGE–KUTTA–LEGENDRE FINITE DIFFERENCE SCHEME

This paper presents the Runge–Kutta–Legendre (RKL) finite difference scheme, allowing for an additional shift in its polynomial representation. A short presentation of the stability region, comparatively to the Runge–Kutta–Chebyshev scheme follows. We then explore the problem of pricing American options with the RKL scheme under the one factor Black–Scholes and the two factor Heston stochastic volatility models, as well as the pricing of butterfly spread and digital options under the uncertain volatility model, where a Hamilton–Jacobi–Bellman partial differential equation needs to be solved. We explore the order of convergence in these problems, as well as the option greeks stability, compared to the literature and popular schemes such as Crank–Nicolson, with Rannacher time-stepping.

Development of time-space adaptive smoothed particle hydrodynamics method with Runge-Kutta Chebyshev scheme

Smoothed particle hydrodynamics (SPH) method is a Lagrangian particle method that has been widely used for solving complex mechanics problems. To reduce the SPH computational cost, the adaptive SPH (ASPH) with time-varying particle distribution has been proposed using the particle splitting and merging techniques. However, the particle splitting in the adaptive SPH results in a decreased time step due to the reduced particle spacing and thus requires an increased amount of computing time. In the present work, an explicit Runge-Kutta Chebyshev time stepping scheme is implemented within the adaptive SPH, by including the stabilizing substeps into the Runge-Kutta integration method with the use of the Chebyshev polynomials. The number of stabilizing substeps in a single SPH time step can be adaptive after particle splitting while the Runge-Kutta integration time step remains the same. Therefore, the proposed SPH algorithm is adaptive in both time and space domains. Representative example problems are simulated using the developed time-space adaptive SPH method. It is demonstrated that, as compared to the standard SPH and the conventional adaptive SPH with varying particle distribution only, the presented time-space ASPH method exhibits much higher efficiency without compromising accuracy and the superiority in terms of stability.

An implicit-explicit Runge-Kutta-Chebyshev finite element method for the nonlinear Lithium-ion battery equations

We introduce a numerical method to integrate the nonlinear system of equations that model the physical-chemical laws of the Lithium-ion batteries. The mathematical model, formulated in Doyle et al. J. Electrochem. Soc. 140 (1993) 1526–1533, is a system of strongly coupled nonlinear parabolic and elliptic equations. Our numerical method combines a second order implicit–explicit Runge–Kutta–Chebyshev scheme for time discretization of the parabolic equations governing the dynamics of the variables ce and cs, with linear finite element space discretization of the system. The implicit-explicit numerical formulation of the parabolic equations allows us to decouple the strong nonlinear reaction terms, which are treated implicitly, from the linear diffusion terms, treated explicitly. This approach is computationally efficient because (a) it has an extended stability region, and (b) the coupled system of algebraic equations becomes a single variable nonlinear equation per mesh point, which is easily solved by Newton method.

Energy-conserving Galerkin approximations for quasigeostrophic dynamics

A method is presented for constructing energy-conserving Galerkin approximations in the vertical coordinate of the full quasigeostrophic model with active surface buoyancy. The derivation generalizes the approach of Rocha et al. (2016) [1] to allow for general bases. Details are then presented for a specific set of bases: Legendre polynomials for potential vorticity and a recombined Legendre basis from Shen (1994) [2] for the streamfunction. The method is tested in the context of linear baroclinic instability calculations, where it is compared to the standard second-order finite-difference method and to a Chebyshev collocation method. The Galerkin scheme is quite accurate even for a small number of degrees of freedom N, and growth rates converge much more quickly with increasing N for the Galerkin scheme than for the finite-difference scheme. The Galerkin scheme is at least as accurate as finite differences and can in some cases achieve the same accuracy as the finite difference scheme with ten times fewer degrees of freedom. The energy-conserving Galerkin scheme is of comparable accuracy to the Chebyshev collocation scheme in most linear stability calculations, but not in the Eady problem where the Chebyshev scheme is significantly more accurate. Finally the three methods are compared in the context of a simplified version of the nonlinear equations: the two-surface model with zero potential vorticity. The Chebyshev scheme is the most accurate, followed by the Galerkin scheme and then the finite difference scheme. All three methods conserve energy with similar accuracy, despite not having any a priori guarantee of energy conservation for the Chebyshev scheme. Further nonlinear tests with non-zero potential vorticity to assess the merits of the methods will be performed in a future work.

An anisotropic adaptive, Lagrange–Galerkin numerical method for spray combustion

We present in this paper a novel numerical method for the simulation of time-dependent, diluted spray combustion problems. Spray combustion is a complex numerical problem owing to the diversity of the phenomena (convection, diffusion, heat transfer, vaporization of droplets and chemical reaction) and the disparity of the scales involved. Our model considers a large number of tiny droplets (liquid phase) surrounded by a gas flow (gas phase), making up a special two-phase system where droplet interaction is neglected on account of the much larger inter-droplet distance compared to the droplet radius typically found in burners. The mathematical formulation of the problem follows a hybrid, Eulerian–Lagrangian approach: the continuous, gas phase is ruled by the conservation equations of mass, momentum, species mass fractions and energy, while the liquid phase is described by a set of ordinary differential equations (ODE) that govern the properties of each droplet along its fluid trajectory. For the resolution of the conservation equations of the gas phase, a Lagrange–Galerkin method in an anisotropic adaptive finite element framework is used. In combination with the aforementioned Lagrange–Galerkin methodology we apply a second order Explicit Runge–Kutta–Chebyshev scheme, also useful to solve the ODE systems of equations for the droplets movement. Explicit Runge–Kutta–Chebyshev preserves numerical stability while also facilitating a decoupled calculation of the unknown variables of the gas and liquid phases, a most beneficial feature from a computationally point of view. Finally, we show several examples to highlight the capabilities of our numerical algorithm in canonical and real-world configurations.

Analysis of a Chebyshev-type pseudo-spectral scheme for the nonlinear Schrödinger equation

In this paper, we derive several error estimates that are pertinent to the study of Chebyshev-type spectral approximations on the real line. The results are applied to construct a stable and accurate pseudo-spectral Chebyshev scheme for the nonlinear Schrödinger equation. The new technique has several computational advantages as compared to Fourier and Hermite-type spectral schemes, described in the literature (see e.g., [1]–[3]. Similar to Hermite-type methods, we do not require domain truncation and/or use of artificial boundary conditions. At the same time, the computational complexity is comparable to the best Fourier-type spectral methods described in the literature.

Analysis of Partition Functions for Metallocenes: Ferrocene, Ruthenocene, and Osmocene.

We present a calculation of the torsional potential of the three metallocenes of the iron group, that is, ferrocene, ruthenocene, and osmocene, calculated with the GAUSSIAN program suite. Both a variational method (through computation of the exact energy levels) and our Chebyshev imaginary time propagation method are used to calculate the hindered rotation partition function, demonstrating the efficiency of the Chebyshev scheme. The transition from a semirigid through a hindered rotor to the free rotor regime is demonstrated, and the effect of the hindered rotation (as opposed to a harmonic) treatment on the thermodynamics of metallocenes is demonstrated.

A Runge–Kutta–Chebyshev SPH algorithm for elastodynamics

A stable and accurate Smoothed Particle Hydrodynamics (SPH) method is proposed for solving elastodynamics in solid mechanics. The SPH method is mesh-free, and it promises to overcome most of disadvantages of the traditional finite element techniques. The absence of a mesh makes the SPH method very attractive for those problems involving large deformations, moving boundaries and crack propagation. However, the conventional SPH method still has significant limitations that prevent its acceptance among researchers and engineers, namely the stability and computational costs. In approximating unsteady problems using the SPH method, attention should be given to the choice of time integration schemes as accuracy and efficiency of the SPH solution may be limited by the timesteps used in the simulation. This study presents an attempt to reconstruct an unconditionally stable SPH method for elastodynamics. To achieve this objective we implement an explicit Runge–Kutta Chebyshev scheme with extended stages in the SPH method. This time stepping scheme adds in a natural way a stabilizing stage to the conventional Runge–Kutta method using the Chebyshev polynomials. Numerical results are shown for several test problems in elastodynamics. For the considered elastic regimes, the obtained results demonstrate the ability of our new algorithm to better maintain the shape of the solution in the presence of shocks.

Chebyshev Collocation in Polymer Field Theory: Application to Wetting Phenomena

An efficient numerical self-consistent field theory (SCFT) algorithm is developed for polymer systems subject to nonperiodic boundary conditions. The method involves collocation with a Chebyshev polynomial basis in order to solve the modified diffusion equations of SCFT and relax the fields experienced by polymer segments to a consistent mean-field solution. For Dirichlet boundary conditions on a block copolymer film, we demonstrate that the Chebyshev scheme preserves spectral accuracy, unlike existing methods based on collocation with a Fourier sine basis or finite differences. This advantage is shown to translate into significant benefits for accuracy, computational cost, and memory requirements in high-resolution SCFT simulations of meso and nanostructured polymer thin films. As a nontrivial application of the new Chebyshev method, the static wetting behavior of a two-dimensional fluid polymer droplet on a solid surface was investigated via SCFT simulations. The interaction between the surface and polymer segments is modeled with a local contact Flory–Huggins parameter that can be mapped to a surface tension. Contact angles obtained from simulations with varying surface and interfacial tensions are shown to be consistent with Young’s equation. The results demonstrate that Chebyshev-based SCFT is a viable and efficient technique for exploring complex morphologies and equilibrium phenomena in polymer thin films.

An essentially non‐oscillatory semi‐Lagrangian method for tidal flow simulations

AbstractWe develop an essentially non‐oscillatory semi‐Lagrangian method for solving two‐dimensional tidal flows. The governing equations are derived from the incompressible Navier–Stokes equations with assumptions of shallow water flows including bed frictions, eddy viscosity, wind shear stresses and Coriolis forces. The method employs the modified method of characteristics to discretize the convective term in a finite element framework. Limiters are incorporated in the method to reconstruct an essentially non‐oscillatory algorithm at minor additional cost. The central idea consists in combining linear and quadratic interpolation procedures using nodes of the finite element where departure points are localized. The resulting semi‐discretized system is then solved by an explicit Runge–Kutta Chebyshev scheme with extended stages. This scheme adds in a natural way a stabilizing stage to the conventional Runge–Kutta method using the Chebyshev polynomials. The proposed method is verified for the recirculation tidal flow in a channel with forward‐facing step. We also apply the method for simulation of tidal flows in the Strait of Gibraltar. In both test problems, the proposed method demonstrates its ability to handle the interaction between water free‐surface and bed frictions. Copyright © 2009 John Wiley & Sons, Ltd.

An Eulerian–Lagrangian method for coupled parabolic-hyperbolic equations

We present an Eulerian–Lagrangian method for the numerical solution of coupled parabolic-hyperbolic equations. The method combines advantages of the modified method of characteristics to accurately solve the hyperbolic equations with an Eulerian method to discretize the parabolic equations. The Runge–Kutta Chebyshev scheme is used for the time integration. The implementation of the proposed method differs from its Eulerian counterpart in the fact that it is applied during each time step, along the characteristic curves rather than in the time direction. The focus is on constructing explicit schemes with a large stability region to solve coupled radiation hydrodynamics models. Numerical results are presented for two test examples in coupled convection-radiation and conduction–radiation problems.

An Implicit-Explicit Runge--Kutta--Chebyshev Scheme for Diffusion-Reaction Equations

An implicit-explicit (IMEX) extension of the explicit Runge--Kutta--Chebyshev (RKC) scheme designed for parabolic PDEs is proposed for diffusion-reaction problems with severely stiff reaction terms. The IMEX scheme treats these reaction terms implicitly and diffusion terms explicitly. Within the setting of linear stability theory, the new IMEX scheme is unconditionally stable for reaction terms having a Jacobian matrix with a real spectrum. For diffusion terms the stability characteristics remain unchanged. A numerical comparison for a stiff, nonlinear radiation-diffusion problem between an RKC solver, an IMEX-RKC solver, and the popular implicit BDF solver VODPK using the Krylov solver GMRES illustrates the excellent performance of the new scheme.

Time propagation and spectral filters in quantum dynamics: A Hermite polynomial perspective

We present an investigation of Hermite polynomials as a basic paradigm for quantum dynamics, and make a thorough comparison with the well-known Chebyshev method. The motivation of the present study is to develop a compact and numerically efficient formulation of the spectral filter problem. In particular, we expand the time evolution operator in a Hermite series and obtain thereby an exponentially convergent propagation scheme. The basic features of the present formulation vı̀s a vı̀s Chebyshev scheme are as follows: (i) Contrary to the Chebyshev scheme Hamiltonian renormalization is not needed. However, an arbitrary time scaling may be necessary in order to avoid numerical hazards, and this time scaling also provides a leverage to accelerate the convergence of the Hermite series. We emphasize the final result is independent of the arbitrary scaling. (ii) As with the Chebyshev scheme the method is of high accuracy but not unitary by definition, and thus any deviation from unitarity may be used as a guideline for accuracy. The calculation of expansion coefficients in the present scheme is extremely simple. To contrast the convergence property of present method with that of the Chebyshev one for finite time propagation, we have introduced a time–energy scaling concept, and this has given rise to a unified picture of the overall convergence behavior. To test the efficacy of the present method, we have computed the transmission probability for a one-dimensional symmetric Eckart barrier, as a function of energy, and shown that the present method, by suitable time–energy scaling, can be very efficient for numerical simulation. Time–energy scaling analysis also suggests that it may be possible to achieve a faster convergence with the Hermite based method for finite time propagation, by a proper choice of scaling parameter. We have further extended the present formulation directed toward the spectral filter problem. In particular, we have utilized the Gaussian damping function for the purpose. The Hermite propagation scheme has allowed all the time integrals to be done fully analytically, a feature not completely shared by the Chebyshev based scheme. As a result, we have obtained a very compact and numerically efficient scheme for the spectral filters to compute the interior eigenspectra of a large rank eigensystem. The present formulation also allows us to obtain a closed form expression to estimate the error of the energies and spectral intensities. As a test, we have utilized the present spectral filter method to compute the highly excited vibrational states for the two-dimensional LiCN (J=0) system and compared with the exact diagonalization result.

Laguerre scheme: Another member for propagating the time-dependent Schrödinger equation

A global propagation scheme for the time-dependent Schr\odinger equation is proposed on the basis of the generating function of the Hermite polynomial for expansion of the evolution operator ${e}^{\ensuremath{-}i\ifmmode \hat{H}\else \^{H}\fi{}t}.$ Theoretical analysis and numerical tests have shown that the present scheme is an equivalent to the Chebyshev scheme with two extra advantages.

Nonadiabatic treatment of molecular systems by the wavepackets method

ABSTRACT = We have already developed the many-electron wavepackets (MEWP) method in order to study the dynamics and electronic structure of molecular systems. We extended the MEWP method to study the nonadiabatic effects and formulated a nonadiabatic molecular theory, where both electron and nucleus are treated equivalently. Then we applied our method to the isotope series of hydrogen molecule i.e., H,, HD, and D,, and calculated the total energy and the average distance between nucleus-nucleus, electron-electron, and nucleus-electron in order to analyze numerically the nonadiabatic effect in the molecule. Finally we calculated the real-time evolution of the polarization by means of Chebyshev scheme; and by Fourier transforming this, we found out the excitation spectrum of the system, which corresponds to the electronic excitation and the nuclear vibrational frequency. 0 1996 John Wiley & Sons, Inc. International Journal of Quantum Chemistry: Quantum Chemistry Symposium 30, 1261 -1270 (1 996) 0 1996 John Wiley & Sons, Inc. CCC 0360-8832 I96 I071 261 -1

Solving the time-dependent Schrödinger equation numerically.

We introduce an explicit scheme to solve the time-dependent Schr\"odinger equation. The scheme is a straightforward extension of the second order differencing scheme to the fourth, sixth, and higher order accuracy. The accuracy is remarkably improved with minor changes in the second order differencing program. This method is conditionally stable. There is a trade-off between the higher order accuracy and the condition of stability. The performance is evaluated and compared to the standard methods, such as the Crank-Nicholson scheme (CN) and the Chebyshev scheme (CH). The new scheme is much more accurate than CN, almost equal to CH.

Time dependent wavepackets and periodic orbit assignment

The Kosloff Chebyshev scheme is used to propagate time dependent wavepackets initiated on the various classical periodic orbits of a coupled Morse oscillator system modelled on the stretching vibrational modes of H2O. It is found that these wavepackets follow parent orbits that are stable for long periods and that the power spectra derived from their autocorrelation functions show simply assignable progressions up to the dissociation limit. By contrast, those initiated on unstable orbits soon break up and, hence, yield more complex power spectra. Eigenvalues extracted from the power spectra by use of an analytically determined lineshape function are in excellent agreement with those determined variationally. Wavefunctions derived from the wavepackets also typically overlap to 99% with their variationally determined counterparts. In addition, an interesting cascade of Fermi resonances associated with highly excited local mode states is identified.