Efficient Spectral Collocation Research Articles

Fractional Adams–Bashforth/Moulton methods: An application to the fractional Keller–Segel chemotaxis system

We first formulate a fractional class of explicit Adams–Bashforth (A-B) and implicit Adams–Moulton (A-M) methods of first- and second-order accuracy for the time-integration of D t τ 0 C u ( x , t ) = g ( t ; u ) , τ ∈ ( 0 , 1 ] , where D t τ 0 C denotes the fractional derivative in the Caputo sense. In this fractional setting and in contrast to the standard Adams methods, an extra history load term emerges and the associated weight coefficients are τ -dependent. However when τ = 1 , the developed schemes reduce to the well-known A-B and A-M methods with standard coefficients. Hence, in terms of scientific computing, our approach constitutes a minimal modification of the existing Adams libraries. Next, we develop an implicit–explicit (IMEX) splitting scheme for linear and nonlinear fractional PDEs of a general advection–reaction–diffusion type, and we apply our scheme to the time–space fractional Keller–Segel chemotaxis system. In this context, we evaluate the nonlinear advection term explicitly, employing the fractional A-B method in the prediction step, and we treat the corresponding diffusion term implicitly in the correction step using the fractional A-M scheme. Moreover, we perform the corresponding spatial discretization by employing an efficient and spectrally-accurate fractional spectral collocation method. Our numerical experiments exhibit the efficiency of the proposed IMEX scheme in solving nonlinear fractional PDEs.

Efficient Spectral Collocation Algorithm for a Two-Sided Space Fractional Boussinesq Equation with Non-local Conditions

A spectral shifted Legendre Gauss–Lobatto collocation method is developed and analyzed to solve numerically one-dimensional two-sided space fractional Boussinesq (SFB) equation with non-classical boundary conditions. The method depends basically on the fact that an expansion in a series of shifted Legendre polynomials $${P_{L,n}(x), \ x\in[0,L]}$$ is assumed, for the function and its space-fractional derivatives occurring in the two-sided SFB equation. The Legendre–Gauss–Lobatto quadrature rule is established to treat the non-local conservation conditions, and then the problem with its non-local conservation conditions is reduced to a system of ordinary differential equations (ODEs) in time. Thereby, the expansion coefficients are then determined by reducing the two-sided SFB with its boundary and initial conditions to a system of ODEs for these coefficients. This system may be solved numerically in a step-by-step manner by using implicit Runge–Kutta method of order four. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented.

A Chebyshev-Gauss Spectral Collocation Method for Odrinary Differential Equations

In this paper, we introduce an efficient Chebyshev-Gauss spectral collocation method for initial value problems of ordinary differential equations. We first propose a single interval method and analyze its convergence. We then develop a multi-interval method. The suggested algorithms enjoy spectral accuracy and can be implemented in stable and efficient manners. Some numerical comparisons with some popular methods are given to demonstrate the effectiveness of this approach.

An efficient collocation algorithm for multidimensional wave type equations with nonlocal conservation conditions

In this paper, we derive and analyze an efficient spectral collocation algorithm to solve numerically some wave equations subject to initial-boundary nonlocal conservation conditions in one and two space dimensions. The Legendre pseudospectral approximation is investigated for spatial approximation of the wave equations. The Legendre–Gauss–Lobatto quadrature rule is established to treat the nonlocal conservation conditions, and then the problem with its nonlocal conservation conditions are reduced to a system of ODEs in time. As a theoretical result, we study the convergence of the solution for the one-dimensional case. In addition, the proposed method is extended successfully to the two-dimensional case. Several numerical examples with comparisons are given. The computational results indicate that the proposed method is more accurate than finite difference method, the method of lines and spline collocation approach.

On cost function transformations for the reduction of uncertain model parameters’ impact towards the optimal solutions

Uncertainties affect the accuracy of nonlinear static or dynamic optimization and inverse problems. The propagation of uncertain model parameters towards the optimal problem solutions can be assessed in a deterministic or stochastic way using Monte Carlo based techniques and efficient spectral collocation and Galerkin projection methods. This paper presents cost function transformations for reducing the impact of uncertain model parameters towards the optimal solutions. We assess the consistency of the methodology by determining sufficient conditions on the cost function transformations and apply the methodology on several test functions.

An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system

In this paper, we derive an efficient spectral collocation algorithm to solve numerically the nonlinear complex generalized Zakharov system (GZS) subject to initial-boundary conditions. The Jacobi pseudospectral approximation is investigated for spatial approximation of the GZS. It possesses the spectral accuracy in space. The Jacobi–Gauss–Lobatto quadrature rule is established to treat the boundary conditions, and then the problem with its boundary conditions is reduced to a system of ordinary differential equations in time variable. This scheme has the advantage of allowing us to obtain the spectral solution in terms of the Jacobi parameters α and β, which therefore means that the algorithm holds a number of collocation methods as special cases. Finally, two illustrative examples are implemented to assess the efficiency and high accuracy of the Jacobi pseudo-spectral scheme.

A shifted Jacobi collocation algorithm for wave type equations with non-local conservation conditions

Abstract In this paper, we propose an efficient spectral collocation algorithm to solve numerically wave type equations subject to initial, boundary and non-local conservation conditions. The shifted Jacobi pseudospectral approximation is investigated for the discretization of the spatial variable of such equations. It possesses spectral accuracy in the spatial variable. The shifted Jacobi-Gauss-Lobatto (SJ-GL) quadrature rule is established for treating the non-local conservation conditions, and then the problem with its initial and non-local boundary conditions are reduced to a system of second-order ordinary differential equations in temporal variable. This system is solved by two-stage forth-order A-stable implicit RK scheme. Five numerical examples with comparisons are given. The computational results demonstrate that the proposed algorithm is more accurate than finite difference method, method of lines and spline collocation approach

An efficient spectral collocation algorithm for nonlinear Phi-four equations

A Jacobi-Gauss-Lobatto collocation method is developed in this work to obtain spectral solutions for different versions of nonlinear time-dependent Phi-four equations subject to nonhomogeneous initial-boundary conditions. The node points are introduced as the roots of the orthogonal Jacobi polynomial with general parameters, α and β. The objective of this paper is thus to investigate the influence of the Jacobi spectral collocation method for solving the nonlinear Phi-four equations. Moreover, the results obtained with the different Jacobi polynomial parameters, α and β are compared to examine the accuracy of most of these parameters. The accuracy and performance of the proposed method are assessed and evaluated through solving three nonlinear problems. Some numerical experiments are presented to show the convergence and the accuracy of the proposed algorithm.

New velocity-space discretization for continuum kinetic calculations and Fokker–Planck collisions

Numerical techniques for discretization of velocity space in continuum kinetic calculations are described. An efficient spectral collocation method is developed for the speed coordinate – the radius in velocity space – employing a novel set of non-classical orthogonal polynomials. For problems in which Fokker–Planck collisions are included, a common situation in plasma physics, a procedure is detailed to accurately and efficiently treat the field term in the collision operator (in the absence of gyrokinetic corrections). When species with disparate masses are included simultaneously, a careful extrapolation of the Rosenbluth potentials is performed. The techniques are demonstrated in several applications, including neoclassical calculations of the bootstrap current and plasma flows in a tokamak.

Efficient spectral collocation and continuation methods for symmetry-breaking solutions of rotating Bose–Einstein condensates

We describe an efficient spectral collocation method (SCM) for symmetry-breaking solutions of rotating Bose–Einstein condensates (BECs) which is governed by the Gross–Pitaevskii equation (GPE). The Lagrange interpolants using the Legendre–Gauss–Lobatto points are used as the basis functions for the trial function space. Some formulas for the derivatives of the basis functions are given so that the GPE can be efficiently computed. The SCMs are incorporated in the context of a predictor–corrector continuation algorithm for tracing primary and secondary solution branches of the GPE. Symmetry-breaking solutions are numerically presented for both rotating BECs, BECs in optical lattices, and two-component BECs in optical lattices. Our numerical results show that the numerical algorithm we propose in this paper outperforms the classical orthogonal Legendre polynomials.

A Two-Parameter Continuation Method for Rotating Two-Component Bose-Einstein Condensates in Optical Lattices

AbstractWe study efficient spectral-collocation and continuation methods (SCCM) for rotating two-component Bose-Einstein condensates (BECs) and rotating two-component BECs in optical lattices, where the second kind Chebyshev polynomials are used as the basis functions for the trial function space. A novel two-parameter continuation algorithm is proposed for computing the ground state and first excited state solutions of the governing Gross-Pitaevskii equations (GPEs), where the classical tangent vector is split into two constraint conditions for the bordered linear systems. Numerical results on rotating two-component BECs and rotating two-component BECs in optical lattices are reported. The results on the former are consistent with the published numerical results.

Three-dimensional linear instability in pressure-driven two-layer channel flow of a Newtonian and a Herschel–Bulkley fluid

The three-dimensional linear stability characteristics of pressure-driven two-layer channel flow are considered, wherein a Newtonian fluid layer overlies a layer of a Herschel–Bulkley fluid. We focus on the parameter ranges for which Squire’s theorem for the two-layer Newtonian problem does not exist. The modified Orr–Sommerfeld and Squire equations in each layer are derived and solved using an efficient spectral collocation method. Our results demonstrate the presence of three-dimensional instabilities for situations where the square root of the viscosity ratio is larger than the thickness ratio of the two layers; these “interfacial” mode instabilities are also present when density stratification is destabilizing. These results may be of particular interest to researchers studying the transient growth and nonlinear stability of two-fluid non-Newtonian flows. We also show that the “shear” modes, which are present at sufficiently large Reynolds numbers, are most unstable to two-dimensional disturbances.

Stability of Plane Channel Flow With Viscous Heating

The linear stability analysis of pressure-driven flow undergoing viscous heating through a channel is considered. The walls of the channel are maintained at different constant temperatures and Nahme’s law is applied to model the temperature dependence of the fluid viscosity. A modified Orr–Sommerfeld equation coupled with a linearized energy equation is derived and solved using an efficient spectral collocation method. Our results indicate that increasing the influence of viscous heating is destabilizing. It is also shown that the critical Reynolds number decreases by one order of magnitude with increase in the Nahme number. An energy analysis is conducted to understand the underlying physical mechanism of the instability.

Linear instability of pressure-driven channel flow of a Newtonian and a Herschel-Bulkley fluid

The linear stability characteristics of pressure-driven two-layer channel flow are considered, wherein a Newtonian fluid layer overlies a layer of a Herschel-Bulkley fluid. A pair of coupled Orr-Sommerfeld eigenvalue equations are derived and solved using an efficient spectral collocation method for cases in which unyielded regions are absent. An asymptotic analysis is also carried out in the long-wave limit, the results of which are in excellent agreement with the numerical predictions. Our analytical and numerical results indicate that increasing the dimensionless yield stress, prior to the formation of unyielded plugs below the interface, is destabilizing. Increasing the shear-thinning tendency of the lower fluid is stabilizing.

An efficient and accurate semivectorial spectral collocation method for analyzing polarized modes of rib waveguides

A novel solution method based on the semivectorial spectral collocation method with domain decomposition is proposed to calculate the modal characteristics of rib waveguides. We formulate the scheme in terms of the transverse magnetic components H/sub x/ and H/sub y/. The optical field expanded by a suitable set of orthogonal basis functions and the refractive index profile are all represented on a grid of discrete points in each subdomain. In addition, the a priori determination of the scaling factor of Laguerre-Gauss (LG) functions is introduced by means of the effective index method. The subdomains are then patched by imposing the continuities of longitudinal electric component E/sub z/ and magnetic component H/sub z/ at all internal dielectric interfaces. Because of the zero divergence constraint of magnetic field vector explicitly included, the occurrence of spurious modes is prevented. The present method is tested for various rib waveguide structures with lossless or lossy materials. Even with a coarse mesh, our results are still found to be in good agreement with those produced by other various full-vectorial methods, but without complexity of the latter approaches.

Unsteady fluid flow and temperature fields in a horizontal enclosure with an adiabatic body

A two-dimensional solution for unsteady natural convection in an enclosure with an adiabatic square body is obtained using an accurate and efficient Chevyshev spectral collocation method. A spectral multi-domain methodology is used to handle an adiabatic body located at the center of computational domain. The physical model considered here is that an adiabatic body is located at the center between the bottom hot and top cold walls. In order to see the effects of the presence of an adiabatic body on time-dependent natural convection between the hot and cold walls, we investigated the detail structure of fluid flow and heat transfer as a function of time for different Rayleigh numbers varying in the range of 103 to 106. When Ra=103, streamlines and isotherms reach the steady state without any oscillatory transients, and the flow and temperature distribution around the body in the enclosure shows a four-fold symmetrical pattern. At Ra=4×103, 104, and 105, streamlines and isotherms reach the steady state after starting oscillatory transients, and the flow and temperature fields change their shape from the symmetrical pattern to the nonsymmetrical one as time goes by. When Ra=106, streamlines and isotherms become time-dependent, and the fluid flow and temperature fields also change their shape in a regularly periodic fashion as a function of time.

TWO-DIMENSIONAL AND UNSTEADY NATURAL CONVECTION IN A HORIZONTAL ENCLOSURE WITH A SQUARE BODY

A two-dimensional solution for unsteady natural convection in an enclosure with a square body is obtained using an accurate and efficient Chevyshev spectral collocation method. A spectral multidomain methodology is used to handle a square body located at the center of the computational domain. The physical model considered here is that a square body is located at the center between the bottom hot and top cold walls. To see the effects of the presence of a body on natural convection between the hot and cold walls, we considered the cases that the body maintains the adiabatic and isothermal thermal boundary conditions for different Rayleigh numbers varying in the range of 103 to 106. When the Rayleigh number is small, the flow and temperature distribution between the hot and cold walls shows a symmetrical and steady pattern. At the intermediate Rayleigh number, the fluid flow and temperature fields maintain the steady state but change their shape to the nonsymmetrical pattern. When the Rayleigh number is high, the flow and temperature fields become time dependent, and their time-averaged shapes approach the symmetric pattern again. The Rayleigh number for the fluid flow and temperature fields to become nonsymmetrical and time dependent depends on the thermal boundary conditions of a body. The variation of time- and surface-averaged Nusselt numbers on the hot and cold walls and at the body surfaces for different Rayleigh numbers and thermal boundary conditions are also presented to show the overall heat transfer characteristics in the system.

Direct Numerical Simulation of Flow and Heat Transfer From a Sphere in a Uniform Cross-Flow

Direct numerical solution for flow and heat transfer past a sphere in a uniform flow is obtained using an accurate and efficient Fourier-Chebyshev spectral collocation method for Reynolds numbers up to 500. We investigate the flow and temperature fields over a range of Reynolds numbers, showing steady and axisymmetric flow when the Reynolds number is less than 210, steady and nonaxisymmetric flow without vortex shedding when the Reynolds number is between 210 and 270, and unsteady three-dimensional flow with vortex shedding when the Reynolds number is above 270. Results from three-dimensional simulation are compared with the corresponding axisymmetric simulations for Re>210 in order to see the effect of unsteadiness and three-dimensionality on heat transfer past a sphere. The local Nusselt number distribution obtained from the 3D simulation shows big differences in the wake region compared with axisymmetric one, when there exists strong vortex shedding in the wake. But the differences in surface-average Nusselt number between axisymmetric and three-dimensional simulations are small owing to the smaller surface area associated with the base region. The shedding process is observed to be dominantly one-sided and as a result axisymmetry of the surface heat transfer is broken even after a time-average. The one-sided shedding also results in a time-averaged mean lift force on the sphere.