The Tau Method as an analytic tool in the discussion of equivalence results across numerical methods

A Tau Method approximate solution of a given differential equation defined on a compact [a, b] is obtained by adding to the right hand side of the equation a specific minimal polynomial perturbation termH _n(x), which plays the role of a representation of zero in [a,b] by elements of a given subspace of polynomials. Neither discretization nor orthogonality are involved in this process of approximation. However, there are interesting relations between the Tau Method and approximation methods based on the former techniques. In this paper we use equivalence results for collocation and the Tau Method, contributed recently by the authors together with classical results in the literature, to identify precisely the perturbation termH(x) which would generate a Tau Method approximate solution, identical to that generated by some specific discrete methods over a given mesh Π ∈ [a, b]. Finally, we discuss a technique which solves the inverse problem, that is, to find adiscrete perturbed Runge-Kutta scheme which would simulate a prescribed Tau Method. We have chosen, as an example, a Tau Method which recovers the same approximation as an orthogonal expansion method. In this way we close the diagram defined by finite difference methods, collocation schemes, spectral techniques and the Tau Method through a systematic use of the latter as an analytical tool.     

Chebyshev polynomial and heat conduction equation in cylindrical geometry

The solutions of the unsteady heat conduction equations in cylindrical geometry in one and two dimensions are obtained using the Chebyshev polynomial expansions in the spatial domain. Equations are discretized in the time domain using the trapezoidal rule. The resulting differential equations are reduced to backward recurrence relations for the coefficients occurring in the Chebyshev polynomial expansions, which are then solved using the Tau method. It is shown that the Chebyshev polynomial solutions produce results to the machine-precision accuracy in the spatial domain using only a modest number of terms, and are, therefore, excellent alternatives to the other techniques used.     

A new second-order accurate time discretization method for the nonlinear cubic Schrödinger equation

A new second-order accurate semi-analytical time discretization method is introduced for the numerical solution of the one-dimensional nonlinear cubic Schrödinger equation. This method is based on the combination of the method of lines, Crank–Nicolson method, Newton method and Lanczos’ Tau method. It is a self-starting averaged two-time-level scheme that has proved to be stable, accurate and energy conservative for long time integration periods. At each time level, approximate solutions are sought on a segmented spatial interval as finite expansions in terms of a given orthogonal polynomial basis mapped appropriately onto each spatial subsegment. We have carried out numerical simulation concerning several cases for the propagation, collision and the bound states of solitons. Accurate results have been obtained using Chebyshev and Legendre polynomials. These results are well comparable with other published results obtained by the use of various standard numerical methods.     

A recursive formulation of Galerkin''s method in terms of the tau method: Bounded and unbounded domains

The recursive formulation of the Tau Method (see [1,2]) is used to simulate the method of Galerkin. As a result of this, a new recursive formulation of Galerkin's Method is deduced. Furthermore, it is shown that the Galerkin's Method is a special form of a Tau Method with a weighted polynomial basis. Examples of application to the numerical approximation of linear and nonlinear partial differential equations, defined on bounded and unbounded domains, are also given.     

Step by step Tau method—Part I. Piecewise polynomial approximations

Lánczos remarked that approximations obtained with the Tau method using a Legendre polynomial perturbation term defined in a finite interval J, give accurate estimations at the end point of J. This fact, coupled with a recursive technique for the generation of Tau approximations described by the author elsewhere (Ortiz, 1969, 1974), is used to construct a step by step formulation of the Tau method in which the error is minimized at the matching point of successive steps. This formulation is applied to the construction of accurate piecewise polynomial approximations with an almost equioscillant error and various degrees of smoothness at the breaking points.

A technique based on the mapping of a master element Tau approximation defined over a finite interval of variable length is used in order to simplify the computational process.

Numerical examples and an estimation of the step size in relation to the size of the error in the equation are also discussed.     

The Variable Order Fast Multipole Method for Boundary Integral Equations of the Second Kind

We discuss the variable order Fast Multipole Method (FMM) applied to piecewise constant Galerkin discretizations of boundary integral equations. In this version of the FMM low-order expansions are employed in the finest level and orders are increased in the coarser levels. Two versions will be discussed, the first version computes exact moments, the second is based on approximated moments. When applied to integral equations of the second kind, both versions retain the asymptotic error of the direct method. The complexity estimate of the first version contains a logarithmic term while the second version is O(N) where N is the number of panels.This work was supported by the NSF under contract DMS-0074553     

An application of domain decomposition methods with non-conforming spectral element/Fourier expansions for the incompressible Navier–Stokes equations

An application of domain decomposition methods is presented for the incompressible Navier–Stokes equations. Non-conforming spectral element/Fourier expansions in the separate domains are employed, and a simple iterative algorithm is used, based on the Dirichlet/Neumann method at the domain interface. Thus, a new element in the present approach is that patching of mixed algebraic/trigonometric polynomial spaces is applied at the domain interface, whereas usually in domain decomposition methods finite-order polynomial spaces have been employed in the separate domains. By applying a coupled scheme for the velocity and pressure fields in each domain, a stable algorithm is obtained. New numerical results are presented on a 3-D model problem of flow in a channel, where both bounding surfaces have corrugations, but of different orientation. Smooth solutions across the domain interface are obtained. Steady flow and the onset of flow instability is simulated and discussed. The results demonstrate that spectral element/Fourier expansions, which have been previously used to study flows in geometries with one homogeneous dimension, may be employed to tackle flow problems in relatively more complex geometries. Furthermore, the results suggest that decomposition into domains with 3-D elements and domains with 2-D elements/Fourier expansions, or domains handled by spectral methods, may be an attractive possibility. The advantage is due to the orthogonality and decoupling of the Fourier modes, which leads to a computational load increasing only linearly with resolution. A related attractive feature is that a natural way for parallel implementation is offered.     

A comparison of the method of lines to finite difference techniques in solving time-dependent partial differential equations

Steady state solutions to two time dependent partial differential systems have been obtained by the Method of Lines (MOL) and compared to those obtained by efficient standard finite difference methods: (i) Burger's equation over a finite space domain by a forward time—central space explicit method, and (ii) the stream function—vorticity form of viscous incompressible fluid flow in a square cavity by an alternating direction implicit (ADI) method. The standard techniques were far more computationally efficient when applicable. In the second example, converged solutions at very high Reynolds numbers were obtained by MOL, whereas solution by ADI was either unattainable or impractical. With regard to “set up” time, solution by MOL is an attractive alternative to techniques with complicated algorithms, as much of the programming difficulty is eliminated.     

Numerical solution of differential eigenvalue problems with an operational approach to the Tau method

A technique for the numerical solution of eigenvalue problems defined by differential equations, based on an operational approach to the Tau method recently proposed by the authors, is shown to be equivalent to a method of Chaves and Oritz. The technique discussed here leads to an algorithmic formulation of remarkable simplicity and to numerical results of high accuracy. It requires no shooting and can deal with complex multipoint boundary conditions and a nonlinear dependence on the eigenvalue parameter.     

Tau-lines: A new hybrid approach to the numerical treatment of crack problems based on the Tau method

A new hybrid computational technique based on Ortiz' recursive formulation of the Tau method is introduced in this paper and applied to some model singular boundary value problems which are relevant to fracture mechanics (modes I and III). This technique, which we call Tau-lines, combines the method of lines with the Tau method. The former is used in the construction of a system of coupled ordinary differential equations which is the discretized model of a given partial differential equation; the latter is used to find an accurate approximation of the solution of such a system which involves no further discretization.Recent theoretical results on the Tau method show that its error is optimal in the sense that, for a given degree n, it has the same order of error as the best uniform approximation of the exact solution by algebraic polynomials of degree n.The present work may be considered as an encouraging first step towards the development of the Tau-lines approach into a useful and efficient computational tool for the numerical treatment of problems in fracture mechanics.     

Reinforcement Learning Based Controller Synthesis for Flexible Aircraft Wings

Aeroelastic study of flight vehicles has been a subject of great interest and research in the last several years. Aileron reversal and flutter related problems are due in part to the elasticity of a typical airplane. Structural dynamics of an aircraft wing due to its aeroelastic nature are characterized by partial differential equations. Controller design for these systems is very complex as compared to lumped parameter systems defined by ordinary differential equations. In this paper, a stabilizing statefeedback controller design approach is presented for the heave dynamics of a wing-fuselage model. In this study, a continuous actuator in the spatial domain is assumed. A control methodology is developed by combining the technique of "proper orthogonal decomposition" and approximate dynamic programming. The proper orthogonal decomposition technique is used to obtain a low-order nonlinear lumped parameter model of the infinite dimensional system. Then a near optimal controller is designed using the single-network-adaptive-critic technique. Furthermore, to add robustness to the nominal single-network-adaptive-critic controller against matched uncertainties, an identifier based adaptive controller is proposed. Simulation results demonstrate the effectiveness of the single-network-adaptive-critic controller augmented with adaptive controller for infinite dimensional systems.      

Pseudospectral methods based on nonclassical orthogonal polynomials for solving nonlinear variational problems

Two direct pseudospectral methods based on nonclassical orthogonal polynomials are proposed for solving finite-horizon and infinite-horizon variational problems. In the proposed finite-horizon and infinite-horizon methods, the rate variables are approximated by the Nth degree weighted interpolant, using nonclassical Gauss-Lobatto and Gauss points, respectively. Exponential Freud type weights are introduced for both of nonclassical orthogonal polynomials and weighted interpolation. It is shown that the absolute error in weighted interpolation is dependent on the selected weight, and the weight function can be tuned to improve the quality of the approximation. In the finite-horizon scheme, the functional is approximated based on Gauss-Lobatto quadrature rule, thereby reducing the problem to a nonlinear programming one. For infinite-horizon problems, an strictly monotonic transformation is used to map the infinite domain onto a finite interval. We transcribe the transformed problem to a nonlinear programming using Gauss quadrature rule. Numerical examples demonstrate the accuracy of the proposed methods.     

Finite elements using long vectors of the DAP

The Finite Element Method for solving partial differential equations using the long vector mode of the DAP is presented. This work was developed on a 32 × 32 version of the DAP attached to a Perq scientific workstation.

First, the implementation of finite elements using the long vector mode of the DAP is given, followed by the treatment of boundary conditions and the solution of the finite element equations using a parallel conjugate gradient method. Two solution procedures for the parallel conjugate gradient method, first without global matrix assembly and second with global matrix assembly, are presented and their advantages and disadvantages are discussed. Preconditioners for the conjugate gradient method based on iteration methods are also discussed and results include a 1-step point Jacobi preconditioner, a m-step point Jacobi preconditioner and a m-step multi-colour preconditioner. Finally long vector implementations for a larger system which stores multinodes per processor using a sliced mapping technique and domain decomposition are included.     

Fractional Laguerre spectral methods and their applications to fractional differential equations on unbounded domain

In this article, we first introduce a singular fractional Sturm–Liouville problem (SFSLP) on unbounded domain. The associated fractional differential operator is both Weyl and Caputo type. The properties of spectral data for fractional operator on unbounded domain have been investigated. Moreover, it has been shown that the eigenvalues of the singular problem are real-valued and the corresponding eigenfunctions are orthogonal. The analytical eigensolutions of SFSLP are obtained and defined as generalized Laguerre fractional-polynomials. The optimal approximation of such generalized Laguerre fractional-polynomials in suitably weighted Sobolev spaces involving fractional derivatives has been derived. We construct an efficient generalized Laguerre fractional-polynomials-Petrov–Galerkin methods for a class of fractional initial value problems and fractional boundary value problems. As a numerical example, we examine space fractional advection–diffusion equation. Our theoretical results are confirmed by associated numerical results.     

Finite Element Simulation of Three-Dimensional Free-Surface Flow Problems

An adaptive finite element algorithm is described for the stable solution of three-dimensional free-surface-flow problems based primarily on the use of node movement. The algorithm also includes a discrete remeshing procedure which enhances its accuracy and robustness. The spatial discretisation allows an isoparametric piecewise-quadratic approximation of the domain geometry for accurate resolution of the curved free surface. The technique is illustrated through an implementation for surface-tension-dominated viscous flows modelled in terms of the Stokes equations with suitable boundary conditions on the deforming free surface. Two three-dimensional test problems are used to demonstrate the performance of the method: a liquid bridge problem and the formation of a fluid droplet.     

An approach for analyzing the reliability of industrial systems using soft-computing based technique

The purpose of this paper is to present a novel technique for analyzing the behavior of an industrial system by utilizing vague, imprecise, and uncertain data. In this, two important tools namely traditional Lambda–Tau and artificial bee colony algorithm have been used to build a technique named as an artificial bee colony (ABC) algorithm based Lambda–Tau (ABCBLT). In real-life situation, data collected from various resources contains a large amount of uncertainties due to human errors and hence it is not easy to analyze the behavior of such system up to a desired accuracy. If somehow behavior of these systems has been calculated, then they have a high range of uncertainty. For handling this situation, a fuzzy set theory has been used in the analysis and an artificial bee colony has been used for determining their corresponding membership functions. To strengthen the analysis, various reliability parameters, which affects the system performance directly, have been computed in the form of fuzzy membership functions. Sensitivity as well as performance analysis has also been analyzed and their computed results are compared with the existing techniques result. The butter–oil processing plant, a complex repairable industrial system has been taken to demonstrate the approach.     

Solution of mixed boundary value problems with local error bound by the finite element method

A method of analysis using finite element techniques is presented for second order, mixed boundary value problems in the plane. The technique focuses computational effort on specific points in the domain and provides absolute solution error bounds at those points by applying the hypercircle method. Solution error of less than 0.0003% and solution error bounds of ± 0.012% are obtained in sample problems. The solution accuracy is notably superior to what is obtained in the traditional finite element method with equivalent discretization. Two problems are presented to illustrate both the strengths and weaknesses of the method.     

Creating a family of collaborative applications for emergency management in the firefighting sub-domain

Software Product Lines allow creating a set of applications that share a set of common features. This makes software product lines appropriate for implementing a family of software products when each stakeholder has different needs and requirements evolve constantly. In the case of emergency management, firefighters have begun using their own smartphones to collaborate and access information during emergencies. However, each firefighter role requires different information and the firefighters’ requirements are constantly evolving. We propose a well-defined process to help stakeholders in this domain specify the products they require, showing that it is possible to apply this software engineering process to extract collaborative requirements common to a set of applications. To confirm whether it was useful for real software implementation, we defined and implemented two applications for this domain. This paper presents the process used to systematically define the domain model and determine the domain scope, which may be used for other domains. We found the process to be appropriate for identifying features related to the domain and its collaborative aspects. The results are promising; the process allowed us to create two working applications which were positively received by two types of stakeholders.     

Two Perturbation Calculations in Fluid Mechanics using Large-expression Management

Two fluid-flow problems are solved using perturbation expansions, with special emphasis on the reduction of intermediate expression swell. This is done by developing tools in Maple that contribute to the efficient representation and manipulation of large expressions. The tools share a common basis, which is the creation of a hierarchy of representation levels such that expressions located at higher levels are expressed using entries from lower levels. The evaluation of higher-level expressions by the algebra system does not proceed recursively to the lowest level, as would ordinarily be the case, but instead can be directly controlled by the user.The first fluid-flow problem, arising in lubrication theory, is solved by implementing a technique of switch-controlled evaluation. The processes of simplification and evaluation are controlled at each level by user-manipulated switches. A perturbation solution is derived semi-interactively with the switch-controlled evaluation being used to reduce the size of intermediate expressions. The second fluid problem, in convection, is solved by extending a perturbation series in several variables to high order by implementing techniques for the automatic generation of hierarchical expression sequences.