Chebyshev Tau Method Research Articles

The performance of a Tau preconditioner on systems of ODEs

In this work, we investigate the new preconditioner of the Tau method, introduced by Ghoreishi and Mohammad Hosseini [13], on systems of ODEs. With a different and relatively straightforward view on the Tau formulation of systems of ODEs this preconditioner is reformulated and explained. An error analysis of the method is also addressed. Some numerical comparisons with the standard Chebyshev-Tau method and the preconditioned method of Cabos confirm the overall efficiency of this preconditioner for systems of ODEs as well.

A spectral method for the wave equation of divergence-free vectors and symmetric tensors inside a sphere

The wave equation for vectors and symmetric tensors in spherical coordinates is studied under the divergence-free constraint. We describe a numerical method, based on the spectral decomposition of vector/tensor components onto spherical harmonics, that allows for the evolution of only those scalar fields which correspond to the divergence-free degrees of freedom of the vector/tensor. The full vector/tensor field is recovered at each time-step from these two (in the vector case), or three (symmetric tensor case) scalar fields, through the solution of a first-order system of ordinary differential equations (ODE) for each spherical harmonic. The correspondence with the poloidal–toroidal decomposition is shown for the vector case. Numerical tests are presented using an explicit Chebyshev-tau method for the radial coordinate.

Numerical solution of hyperbolic telegraph equation using the Chebyshev tau method

AbstractIn this article we propose a numerical scheme to solve the one‐dimensional hyperbolic telegraph equation. The method consists of expanding the required approximate solution as the elements of shifted Chebyshev polynomials. Using the operational matrices of integral and derivative, we reduce the problem to a set of linear algebraic equations. Some numerical examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces very accurate results. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010

Influence of Rayleigh effect combined with Marangoni effect on the onset of convection in a liquid layer overlying a porous layer

The coupling mechanism of Rayleigh effect and Marangoni effect in a liquid–porous system is investigated using a linear stability analysis. The eigenvalue problem is solved by means of a Chebyshev tau method. Results indicate that there are three coupling modes between the Rayleigh effect and the Marangoni effect for different depth ratios.

A Non-axisymmetric Spherical α 2 -Dynamo

Using the Chebyshev-tau method, the generation of oscillatory non-axisymmetric stellar magnetic fields by the α 2 -dynamo is studied in spherical geometry. Following the boundary conditions given by Schubert & Zhang, the spherical α 2 -dynamo consists of a fully convective spherical shell with inner radius r i and outer radius r o . A comparison of the critical dynamo numbers of axisymmetric and -dependent modes for different thicknesses of the convective shell and different α -profiles leads to the following qualitative results: (i) when the angular factor of α -profile is sin n θ cos θ ( n = 1,2,4) the solutions of the α 2 -dynamo are oscillatory and non-axisymmetric, (ii) the thinner the convective shell, the more easily is the non-axisymmetric mode excited and the higher is the latitudinal wave number, (iii) the thickness of the outer convective shell has an effect on the symmetries of the magnetic fields.

A Legendre spectral element method for eigenvalues in hydrodynamic stability

A Legendre polynomial-based spectral technique is developed to be applicable to solving eigenvalue problems which arise in linear and nonlinear stability questions in porous media, and other areas of Continuum Mechanics. The matrices produced in the corresponding generalised eigenvalue problem are sparse, reducing the computational and storage costs, where the superimposition of boundary conditions is not needed due to the structure of the method. Several eigenvalue problems are solved using both the Legendre polynomial-based and Chebyshev tau techniques. In each example, the Legendre polynomial-based spectral technique converges to the required accuracy utilising less polynomials than the Chebyshev tau method, and with much greater computational efficiency.

Numerical piecewise approximate solution of Fredholm integro-differential equations by the Tau method

A general form of numerical piecewise approximate solution of linear integro-differential equations of Fredholm type is discussed. It is formulated for using the operational Tau method to convert the differential part of a given integro-differential equation, or IDE for short, to its matrix representation. This formulation of the Tau method can be useful for such problems over long intervals and also can be used as a good and simple alternative algorithm for other piecewise approximations such as splines or collocation. A Tau error estimator is also adapted for piecewise application of the Tau method. Some numerical examples are considered to demonstrate the implementation and general effect of application of this (segmented) piecewise Chebyshev Tau method.

Numerical approximation of the spectra of Phan-Thien Tanner liquids

The linear stability of the linear Phan-Thien Tanner (PTT) fluid model is investigated for plane Poiseuille flow. The PTT model involves parameters that can be used to fit shear and extensional data, which makes it suitable for describing both polymer solutions and melts. The base flow is determined using a Chebyshev-tau method. The linear stability equations are also discretized using Chebyshev approximations to furnish a generalized eigenvalue problem. The spectrum is shown to comprise a continuous part and a discrete part. The theoretical and numerical results are validated for the UCM and Oldroyd-B models, which are special cases of the PTT model, by comparing with results in the literature. It is demonstrated that the linear extensional and elasticity parameters considered. The computational efficiency and accuracy of the numerical method are also investigated.

Direct numerical simulation of compressible turbulent channel flow between adiabatic and isothermal walls

In this paper, the effects of adiabatic and isothermal conditions on the statistics in compressible turbulent channel flow are investigated using direct numerical simulation (DNS). DNS of two compressible turbulent channel flows (Cases 1 and 2) are performed using a mixed Fourier Galerkin and B-spline collocation method. Case 1 is compressible turbulent channel flow between isothermal walls, which corresponds to DNS performed by Coleman et al. (1995). Case 2 is the flow between adiabatic and isothermal walls. The flow of Case 2 can be a very useful framework for the present objective, since it is the simplest turbulent channel flow with an adiabatic wall and provides ideal information for modelling the compressible turbulent flow near the adiabatic wall. Note that compressible turbulent channel flow between adiabatic walls is not stationary if there is no sink of heat. In Cases 1 and 2, the Mach number based on the bulk velocity and sound speed at the isothermal wall is 1.5, and the Reynolds number based on the bulk density, bulk velocity, channel half-width, and viscosity at the isothermal wall is 3000.To compare compressible and incompressible turbulent flows, DNS of two incompressible turbulent channel flows with passive scalar transport (Cases A and B) are performed using a mixed Fourier Galerkin and Chebyshev tau method. The wall boundary conditions of Cases A and B correspond to those of Cases 1 and 2, respectively. Case A corresponds to the DNS of Kim & Moin (1989). In Cases A and B, the Reynolds number based on the friction velocity, the channel half-width, and the kinematic viscosity is 150.The mean velocity and temperature near adiabatic and isothermal walls for compressible turbulent channel flow can be explained using the non-dimensional heat flux and the friction Mach number. It is found that Morkovin's hypothesis is not applicable to the near-wall asymptotic behaviour of the wall-normal turbulence intensity even if the variable property effect is taken into account. The mechanism of the energy transfers among the internal energy, mean and turbulent kinetic energiesis investigated, and the difference between the energy transfers near isothermal and adiabatic walls is revealed. Morkovin's hypothesis is not applicable to the correlation coeffcient between velocity and temperature fluctuations near the adiabatic wall.

Rational Chebyshev tau method for solving higher-order ordinary differential equations

An approximate method for solving higher-order ordinary differential equations is proposed. The approach is based on a rational Chebyshev (RC) tau method. The operational matrices of the derivative and product of RC functions are presented. These matrices together with the tau method are utilized to reduce the solution of the higher-order ordinary differential equations to the solution of a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

The Tau method and a new preconditioner

In relations to the order of linear ordinary differential equations, using a modified form of the Chebyshev or Legendre and Gegenbauer polynomials, some particular integral operators are introduced. These are used to give a factorization of the operators arising from the application of the Chebyshev or Legendre Tau method. The New-Tau method presented in this article is then compared with the standard Tau method and preconditioned method of Cabos. The New-Tau method shows a superior performance. An analysis of error and a bound for condition number is given. Numerical examples applying iterative solvers show dramatic reduction in condition number and improved convergence for the Tau method with the new preconditioner.

Rational Chebyshev tau method for solving Volterra’s population model

An approximate method for solving Volterra’s population model for population growth of a species in a closed system is proposed. Volterra’s model is a nonlinear integro-differential equation where the integral term represents the effect of toxin. The approach is based on a rational Chebyshev tau method. The Volterra’s population model is first converted to a nonlinear ordinary differential equation. The operational matrices of derivative and product of rational Chebyshev functions are presented. These matrices together with the tau method are then utilized to reduce the solution of the Volterra’s model to the solution of a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique and a comparison is made with existing results.

Hydrodynamic stability, the Chebyshev tau method and spurious eigenvalues

The Chebyshev tau method is examined; a numerical technique which in recent years has been successfully applied to many hydrodynamic stability problems. The orthogonality of Chebyshev functions is used to rewrite the differential equations as a generalized eigenvalue problem. Although a very efficient technique, the occurrence of spurious eigenvalues, which are not always easy to identify, may lead one to believe that a system is unstable when it is not. Thus, the elimination of spurious eigenvalues is of great importance. Boundary conditions are included as rows in the matrices of the generalized eigenvalue problem and these have been observed to be one cause of spurious eigenvalues. Removing boundary condition rows can be difficult. This problem is addressed here, in application to the Benard convection problem, and to the Orr-Sommerfeld equation which describes parallel flow. The procedure given here can be applied to a wide range of hydrodynamic stability problems.

Continuum Mechanics and Applications in Geophysics and the Environment

I Applied Continuum Mechanics.- Numerical Investigation of Shock Waves in a Radiating Gas Described by a Variable Eddington Factor.- Anisotropic Fluids: From Liquid Crystals to Granular Materials.- Integration and Segregation in a Population - A Thermodynamicists's View.- Asymptotic and Other Properties of Some Nonlinear Diffusion Models.- The Binary Mixtures of Euler Fluids: A Unified Theory of Second Sound Phenomena.- Continuously Distributed Control of Plates by Electric Networks with PZT Actuators.- II Soil Mechanics and Porous Media.- Hydraulic Theory for a Frictional Debris Flow on a Collisional Shear Layer.- The Beavers and Joseph Condition for Velocity Slip at the Surface of a Porous Medium.- Porous Convection, the Chebyshev Tau Method, and Spurious Eigenvalues.- Mechanics of Multiphase Porous Media - Application to Unsaturated Soils.- III Glacier and Ice Dynamics.- Modelling Iceberg Drift and Ice-Rafted Sedimentation.- Modelling the Flow of Glaciers and Ice Sheets.- Notes on Basic Glaciological Computational Methods and Algorithms.- Constitutive Modelling and Flow Simulation of Anisotropic Polar Ice.- Influence of Bed Topography on Steady Plane Ice Sheet Flow.- IV Climatology and Lake Physics.- Glacial Isostasy: Models for the Response of the Earth to Varying Ice Loads.- Arctic Sea Ice and Its Role in Climate Variability and Change.- The Role of Simple Models in Understanding Climate Change.- Comparing Different Numerical Treatments of Advection Terms for Wind-Induced Circulations in Lake Constance.

Chebyshev expansion method for solving second and fourth-order elliptic equations

In this paper we introduce a new spectral method based on Chebyshev polynomials for solving second and fourth-order elliptic equations. Moreover the suggested method is applicable for a wide area of differential equations. An explicit formula for the Chebyshev polynomials in terms of arbitrary order of their derivatives is presented. Also a formula for the successive integration of Chebyshev polynomials in terms of Chebyshev polynomials is proved. Numerical results indicate that the suggested method is significantly more accurate than that based on the Chebyshev–Tau method. The present results are in satisfactory agreement with the exact solutions.

Surface-Tension-Driven Convection in a Fluid Overlying a Porous Layer

An accurate numerical analysis for the onset of thermal convection in a two-layer system is presented. The system comprises a saturated porous layer over which lies a layer of the same fluid. The layered system is heated from below, the upper (fluid) surface is free to the atmosphere, and convection driven by surface tension is allowed for. The eigenvalues and eigenfunctions for the instability problem are derived by utilizing a D2 Chebyshev tau method (J. J. Dongarra, B. Straughan, and D. W. Walker, 1996, Appl. Numer. Math.22, 399–435). This allows us to obtain highly accurate eigenvalues and eigenfunctions in a very efficient manner. The onset of convection is seen to have a bimodal nature in which convection may be dominated by the porous medium or by the fluid, depending on the depths of the relative layers and the strength of the tension in the fluid surface. The effect of surface tension is investigated in detail and it is found that for the parameter d̂ (=depth of fluid layer/depth of porous layer) very small, the surface tension has a strong effect on convection dominated by the porous medium, whereas for d̂ larger the surface tension effect is observed only with the fluid mode.

A comparison of the Graffi and Kazhikhov–Smagulov models for top heavy pollution instability

A model to describe convective overturning of a fluid layer due to density differences is derived based on equations of Kazhikhov & Smagulov. This is related to an analogous model of a reduced system based on equations of Dario Graffi. It is shown how the Graffi equations are recovered from the Kazhikhov–Smagulov equations as a non-dimensional parameter G , the Graffi number, tends to zero. The model is analysed numerically and instability thresholds are derived. It is seen that the results are realistic for small diffusion but for relatively large diffusion the approximation of Kazhikhov and Smagulov may have to be replaced by the full non-linear version. The question of spurious eigenvalues is addressed in two versions of the Chebyshev tau method employed in the numerical solution of the instability problem. It is seen that for the Kazhikhov–Smagulov theory the question of spurious eigenvalues is a non-trivial one.

An accurate solution of the Poisson equation by the Chebyshev-Tau method

A new Tau method is presented for the two-dimensional Poisson equation. Comparison of the results for the test problem u( x, y) = sin(4 πx) sin(4 πy) with those computed by Haidvogel and Zang, using the matrix diagonalization method, and Dang-Vu and Delcarte, using the Chebyshev collocation method, indicates that our method would be more accurate.

Multi-component diffusion and penetrative convection

The problem is the investigation of penetrative thermal convection in a three constituent fluid. The stability criteria are derived by a Chebyshev tau method. Very interesting neutral curves are derived including non-perfect “heart shaped” oscillatory curves affording a variety of complex dynamical behaviour, and an isolatory curve allowing stability in distinct finite regions of the Rayleigh number space. When the penetration effect is increased the closed oscillatory curves become more skew and for certain parameters it is found that the closed curve has the same minimum critical Rayleigh number as the unbounded stationary convection one. This leads to the novel prediction that different types of instability will occur at the same critical Rayleigh number but for different wave numbers. The behaviour of the growth rate is investigated to some extent and we see that an exchange of stabilities is occurring between the leading group of eigenvalues (i.e. the growth rate) whereby the eigenvalue with greatest real part may be a complex conjugate pair which is “overtaken” by a purely real eigenvalue in a suitable part of the Rayleigh number domain.

Chebyshev tau-QZ algorithm methods for calculating spectra of hydrodynamic stability problems

The Chebyshev tau method is examined in detail for a variety of eigenvalue problems arising in hydrodynamic stability studies, particularly those of Orr-Sommerfeld type. We concentrate on determining the whole of the top end of the spectrum in parameter ranges beyond those often explored. The method employing a Chebyshev representation of the fourth derivative operator, D 4, is compared with those involving the second and first derivative operators, D 2 and D, respectively. The latter two representations require use of the QZ algorithm in the resolution of the singular generalised matrix eigenvalue problem which arises. Physical problems explored are those of Poiseuille flow, Couette flow, pressure gradient driven circular pipe flow, and Couette and Poiseuille problems for two viscous, immiscible fluids, one overlying the other.