Spectral Differentiation Matrix Research Articles

Highly accurate calculation of higher energy eigenvalues for the radial Schrödinger eigenproblems

In this paper, we derive an efficient numerical scheme for approximating the energy eigenvalues and corresponding wavefunctions of the radial Schrödinger eigenproblems defined on a semi-infinite domain for arbitrary values of the angular momentum number. The numerical scheme is based on the Chebyshev spectral collocation differentiation matrix method. In this scheme, at first, we redefine the radial Schrödinger eigenproblem on a finite interval by adopting an appropriate change of the independent variable. Then, by expanding the wavefunctions of the problem as a series of the Chebyshev polynomials of the first kind, as well as employing the differentiation matrices in order to determine the derivatives of Chebyshev polynomials at the collocation points, considered problem becomes a generalized eigenvalue problem. The convergence behavior and excellent performance of the proposed technique are illustrated through some numerical experiments of the most significant potentials in quantum mechanics. Compared with exact solutions (when available), and those reported previously in the literature, the current scheme achieves higher accuracy and efficiency even when high-index energy eigenvalues are computed.

Convective Heat Transfer of a Hybrid Nanofluid over a Nonlinearly Stretching Surface with Radiation Effect

The flow of the hybrid nanofluid (copper–titanium dioxide/water) over a nonlinearly stretching surface was studied with suction and radiation effect. The governing partial differential equations were then converted into non-linear ordinary differential equations by using proper similarity transformations. Therefore, these equations were solved by applying a numerical technique, namely Chebyshev pseudo spectral differentiation matrix. The results of the flow field, temperature distribution, reduced skin friction coefficient and reduced Nusselt number were deduced. It was found that the rising of the mass flux parameter slows down the velocity and, hence, decreases the temperature. Further, on enlarging the stretching parameter, the velocity and temperature increases and decreases, respectively. In addition, it was mentioned that the radiation parameter can effectively control the thermal boundary layer. Finally, the temperature decreases when the values of the temperature parameter increases.

Radiation and Mixed Convection Magnetohydrodynamics Boundary Layer of Hybrid Cu–Al2O3/Water Nanofluid Flow Over a Wall Jet

The mixed convection MHD boundary layer flow of the hybrid nanofluid (copper–titanium dioxide/water) over a permeable wall jet surface subject to thermal radiation was studied with the effects of suction/injection and thermal energy. The governing PDEs were converted into non-linear ODEs by using the proper similarity transformations. Then, these equations were numerically solved applying the method of Chebyshev pseudo spectral differentiation matrix (ChPDM). The results were deduced for the flow field, temperature distribution, reduced skin friction coefficient and reduced Nusselt number. It was found that the hydrodynamic and thermal boundary layers decreases and increases, respectively, as the strength of the magnetic field increases. Further, in the industrial applications, the normal angle is to be applied if low velocity and high temperature is needed. Moreover, as the mixed convection parameter rises, the velocity and temperature increases and decreases, respectively. In addition, it was mentioned that the radiation parameter can control the thermal boundary layer. Finally, the Cu-nanoparticles volume fraction plays a very important role in the velocity variation.

Chebyshev spectral variational integrator and applications

The Chebyshev spectral variational integrator (CSVI) is presented in this paper. Spectral methods have aroused great interest in approximating numerically a smooth problem for their attractive geometric convergence rates. The geometric numerical methods are praised for their excellent long-time geometric structure-preserving properties. According to the generalized Galerkin framework, we combine two methods together to construct a variational integrator, which captures the merits of both methods. Since the interpolating points of the variational integrator are chosen as the Chebyshev points, the integration of Lagrangian can be approximated by the Clenshaw-Curtis quadrature rule, and the barycentric Lagrange interpolation is presented to substitute for the classic Lagrange interpolation in the approximation of configuration variables and the corresponding derivatives. The numerical float errors of the first-order spectral differentiation matrix can be alleviated by using a trigonometric identity especially when the number of Chebyshev points is large. Furthermore, the spectral variational integrator (SVI) constructed by the Gauss-Legendre quadrature rule and the multi-interval spectral method are carried out to compare with the CSVI, and the interesting kink phenomena for the Clenshaw-Curtis quadrature rule are discovered. The numerical results reveal that the CSVI has an advantage on the computing time over the whole progress and a higher accuracy than the SVI before the kink position. The effectiveness of the proposed method is demonstrated and verified perfectly through the numerical simulations for several classical mechanics examples and the orbital propagation for the planet systems and the Solar system.

A conservative sine pseudo-spectral-difference method for multi-dimensional coupled Gross–Pitaevskii equations

In this paper, a sine pseudo-spectral-difference scheme that preserves the discrete mass and energy is presented and analyzed for the coupled Gross–Pitaevskii equations with Dirichlet boundary conditions in several spatial dimensions. The Crank–Nicolson finite difference method is employed for approximating the time derivative, and the second-order sine spectral differentiation matrix is deduced and applied in spatial discretization. Without any restrictions on the grid ratios, optimal error estimates are established by utilizing the discrete energy method and the equivalence of (semi-)norms. An accelerated algorithm is developed to speed up the numerical implementation with the help of fast sine transform. Numerical examples are tested to confirm the effectiveness and high accuracy of the method.

An efficient spectral-collocation difference method for two-dimensional Schrödinger equation with Neumann boundary conditions

Based on the second-order cosine spectral differentiation matrix deduced in this paper, we present an efficient and high accurate numerical scheme for solving two-dimensional Schrödinger equation with Neumann boundary conditions. The Crank–Nicolson finite difference scheme is utilized in temporal discretization and the cosine spectral-collocation method is employed for the approximation of Laplacian operator. The new scheme conserves the mass and energy in discrete level and is implemented by a fast algorithm in terms of the relations between the cosine spectral differentiation matrix and fast cosine transform. More importantly, this strategy can be extended to address the problems with high (even)-order derivatives in space. Numerical examples and applications are listed to confirm the validity and high accuracy of the method.

Numerical analysis of nonlinear free and forced vibrations of buckled curved beams resting on nonlinear elastic foundations

This paper presents a novel numerical procedure to predict nonlinear free and steady state forced vibrations of clamped–clamped curved beam in the vicinity of postbuckling configuration. Nonlinear Euler–Bernoulli kinematics assumptions including mid-plane stretching are proposed to exhibit a large deformation but a small strain of von Kármán. To simulate the interaction of beam with the surrounding elastic medium, nonlinear elastic foundation with cubic nonlinearity and shearing layer are employed. The nonlinear integro-differential equation that governs the buckling of beam is discretized using the differential-integral quadrature method (DIQM) and then is solved using Newton’s method. The problem of linear vibration is discretized using DIQM and then is solved as a linear eigenvalue problem. Afterwards, a single-mode Galerkin discretization is used to reduce the nonlinear governing equation into a time-varying Duffing equation. The Spectral differentiation matrix operators are exploited to discretize the Duffing equation. The discretized Duffing equation is a nonlinear eigenvalue problem which is directly solved using pseudo arc length continuation method. Results obtained by the proposed numerical solution are compared with analytical solutions available in the literature and good agreement is obtained. Parametric studies are carried out to show the effects of applied axial load, imperfection and nonlinear elastic foundations on the natural frequency as well as forced damped vibration behavior of the beam. The above mention effects play very important role on the dynamic behavior of buckled curved beam.

An extension of the Gegenbauer pseudospectral method for the time fractional Fokker‐Planck equation

The time fractional Fokker‐Planck equation has been used in many physical transport problems which take place under the influence of an external force field. In this paper we examine pseudospectral method based on Gegenbauer polynomials and Chebyshev spectral differentiation matrix to solve numerically a class of initial‐boundary value problems of the time fractional Fokker‐Planck equation on a finite domain. The presented method reduces the main problem to a generalized Sylvester matrix equation, which can be solved by the global generalized minimal residual method. Some numerical experiments are considered to demonstrate the accuracy and the efficiency of the proposed computational procedure.

A conservative Fourier pseudospectral algorithm for the nonlinear Schrödinger equation

In this paper, we derive a new method for a nonlinear Schrödinger system by using the square of the first-order Fourier spectral differentiation matrix D1 instead of the traditional second-order Fourier spectral differentiation matrix D2 to approximate the second derivative. We prove that the proposed method preserves the charge and energy conservation laws exactly. A deduction argument is used to prove that the numerical solution is second-order convergent to the exact solutions in ‖ · ‖2 norm. Some numerical results are reported to illustrate the efficiency of the new scheme in preserving the charge and energy conservation laws.

Multi-Symplectic Fourier Pseudospectral Method for the Kawahara Equation

AbstractIn this paper, we derive a multi-symplectic Fourier pseudospectral scheme for the Kawahara equation with special attention to the relationship between the spectral differentiation matrix and discrete Fourier transform. The relationship is crucial for implementing the scheme efficiently. By using the relationship, we can apply the Fast Fourier transform to solve the Kawahara equation. The effectiveness of the proposed methods will be demonstrated by a number of numerical examples. The numerical results also confirm that the global energy and momentum are well preserved.

On the forced vibration analysis of Timoshenko nanobeams based on the surface stress elasticity theory

This paper aims to present nonlinear forced vibration characteristics of nanobeams including surface stress effect. By considering the local geometrical nonlinearity based on von Karman relation, a new formulation of the Timoshenko beam model is developed through the Gurtin–Murdoch elasticity theory in which the effect of surface stress is incorporated. By using a variational approach on the basis of Hamilton’s principle, the size-dependent equations of motion and associated boundary conditions are obtained. The generalized differential quadrature (GDQ) method is employed to discretize the non-classical governing differential equations over the spatial domain by using the shifted Chebyshev–Gauss–Lobatto grid points. Subsequently, a Galerkin-based numerical approach is put to use in order to reduce the set of nonlinear equations into a time-varying set of ordinary differential equations of Duffing-type. In the next step, the time domain is discretized via spectral differentiation matrix operators which are defined based on the derivatives of a periodic base function. Finally, the pseudo arc-length method is employed to solve the resulting nonlinear parameterized algebraic equations. The frequency–response curves for forced vibration behavior of nanobeams including the effect of surface stress are predicted corresponding to various values of beam thickness, length to thickness ratio and surface elastic constants. It is revealed that by incorporating the surface stress effect, the maximum amplitude occurs at lower excitation frequencies and the wide of region of the response tends to decrease.

A conservative Fourier pseudospectral algorithm for a coupled nonlinear Schrödinger system

We derive a new method for a coupled nonlinear Schrödinger system by using the square of first-order Fourier spectral differentiation matrix D1 instead of traditional second-order Fourier spectral differentiation matrix D2 to approximate the second derivative. We prove the proposed method preserves the charge and energy conservation laws exactly. In numerical tests, we display the accuracy of numerical solution and the role of the nonlinear coupling parameter in cases of soliton collisions. Numerical experiments also exhibit the excellent performance of the method in preserving the charge and energy conservation laws. These numerical results verify that the proposed method is both a charge-preserving and an energy-preserving algorithm.

Seismic imaging based on spectral differentiation matrix and GPU implementation

article i nfo Article history: Finite-difference depth migration based on one-way wave equation uses second-order, fourth-order, or other finite-order approximations for spatial derivatives. These finite-order approximations often lead to spatial dispersion errors and low accuracy. To avoid these errors, smaller mesh spacings are used, which results in huge increase in computation cost. In this paper, we develop a new spectral differentiation matrix method for approximating spatial derivatives. The approximation of spectral differentiation matrix is of infinite- order, and therefore avoids dispersion errors and increases accuracy by using a smaller number of grid points. Recently-developed GPU computational technology is well suited to the seismic migration based on frequency-domain wavefield-continuation methods because such approaches can be sufficiently decoupled. We present a frequency-domain wavefield-continuation algorithm based on spectral differentiation matrix, and realize GPU implementation of this new algorithm on Marmousi model.

Matrix-based Chebyshev spectral approach to dynamic analysis of non-uniform Timoshenko beams

A Chebyshev spectral method (CSM) for the dynamic analysis of non-uniform Timoshenko beams under various boundary conditions and concentrated masses at their ends is proposed. The matrix-based Chebyshev spectral approach was used to construct the spectral differentiation matrix of the governing differential operator and its boundary conditions. A matrix condensation approach is crucially presented to impose boundary conditions involving the homogeneous Cauchy conditions and boundary conditions containing eigenvalues. By taking advantage of the standard powerful algorithms for solving matrix eigenvalue and generalized eigenvalue problems that are embodied in the MATLAB commands, chebfun and eigs, the modal parameters of non-uniform Timoshenko beams under various boundary conditions can be obtained from the eigensolutions of the corresponding linear differential operators. Some numerical examples are presented to compare the results herein with those obtained elsewhere, and to illustrate the accuracy and effectiveness of this method.

Mapped barycentric Chebyshev differentiation matrix method for the solution of regular Sturm–Liouville problems

In this paper, using spectral differentiation matrix and an elimination treatment of boundary conditions, Sturm–Liouville problems (SLPs) are discretized into standard matrix eigenvalue problems. The eigenvalues of the original Sturm–Liouville operator are approximated by the eigenvalues of the corresponding Chebyshev differentiation matrix (CDM). This greatly improves the efficiency of the classical Chebyshev collocation method for SLPs, where a determinant or a generalized matrix eigenvalue problem has to be computed. Furthermore, the state-of-the-art spectral method, which incorporates the barycentric rational interpolation with a conformal map, is used to solve regular SLPs. A much more accurate mapped barycentric Chebyshev differentiation matrix (MBCDM) is obtained to approximate the Sturm–Liouville operator. Compared with many other existing methods, the MBCDM method achieves higher accuracy and efficiency, i.e., it produces fewer outliers. When a large number of eigenvalues need to be computed, the MBCDM method is very competitive. Hundreds of eigenvalues up to more than ten digits accuracy can be computed in several seconds on a personal computer.

Modal expansion of one-way operators based on spectral differentiation matrix

We explore the modal expansion of one-way operators based on the spectral differentiation matrix. The results are compared with the expansions based on the difference matrices with increasing order of accuracy. The modal expansion plus spectral differentiation matrix provides a very accurate discretization of the one-way operators.

A numerical solution of the generalized Burger’s–Huxley equation by spectral collocation method

In this paper, we present a new method for solving of the generalized Burger’s–Huxley equation by using the collocation formula for calculating spectral differentiation matrix for Chebyshev–Gauss–Lobatto points. Firstly, theory of application of pseudospectral method on Burger’s–Huxley equations presented. This method yields Burger’s–Huxley equation to a system of ordinary differential equations (ODEs). Secondly, we use fourth-order Runge–Kutta formula for the numerical integration of the system of ODE. The numerical results obtained by this way have been compared with the exact solution to show the efficiency of the method.

Modified pseudospectral method for generalized Burger's-Fisher equation

We consider solitary-wave solutions of the generalized Burger’s-Fisher equation ∂Ψ ∂t + αΨ δ ∂Ψ ∂x − ∂ 2 Ψ ∂x2 = βΨ(1 − Ψ δ ). In this paper, we present a new method for solving of the generalized Burger’s-Fisher equation by using the collocation formula for calculating spectral differentiation matrix for Chebyshev-Gauss-Lobatto point. To reduce round-off error in spectral collocation method we use a new precondotioning. Firstly, theory of application of spectral collocation method on Burger’s-Fisher equation presented. This method yields Burger’s-Fisher equation to a system of ordinary differential equations(ODEs). Secondly, we use forth order Runge-Kutta formula for the numerical integration of the system of ODE. The numerical results obtained by this way have been compared with the exact solution to show the efficiency of the method.

A numerical solution of the generalized Burger’s–Huxley equation by pseudospectral method and Darvishi’s preconditioning

In this paper, we present a new method for solving of the generalized Burger’s–Huxley equation by using the collocation formula for calculating spectral differentiation matrix for Chebyshev–Gauss–Lobatto point. To reduce round-off error in spectral collocation method we use Darvishi’s precondotioning. Firstly, theory of application of spectral collocation method on Burger’s–Huxley equation presented. This method yields Burger’s–Huxley equation to a system of ordinary differential equations (ODEs). Secondly, we use forth order Runge–Kutta formula for the numerical integration of the system of ODE. The numerical results obtained by this way have been compared with the exact solution to show the efficiency of the method.

Spectral collocation method for the solution of the generalized Burger–Fisher equation

In this paper, we present a new method for solving the generalized Burger–Fisher equation by using the collocation formula for calculating spectral differentiation matrix for Chebyshev–Gauss–Lobatto point. Firstly, theory of application of pseudospectral method on Burger–Fisher equations is presented. This method yields Burger–Fisher equation to a system of ordinary-differential equations (ODEs). Secondly, we use forth-order Runge–Kutta formula for the numerical integration of the system of ODE. The numerical results obtained by this way have been compared with the exact solution to show the efficiency of the method.