Convective Eigenvalue Problems for Convergence Enhancement of Eigenfunction Expansions in Convection–Diffusion Problems

2017 ◽  
Vol 10 (2) ◽  
Author(s):  
Renato M. Cotta ◽  
Carolina P. Naveira-Cotta ◽  
Diego C. Knupp
Keyword(s):  
Eigenvalue Problem ◽  
Eigenfunction Expansion ◽  
Space Variable ◽  
Eigenvalue Problems ◽  
Integral Transform ◽  
Diffusion Problem ◽  
Eigenfunction Expansions ◽  
Diffusion Operator ◽  
Convection Diffusion ◽  
Diffusion Problems

The present work considers the application of the generalized integral transform technique (GITT) in the solution of a class of linear or nonlinear convection–diffusion problems, by fully or partially incorporating the convective effects into the chosen eigenvalue problem that forms the basis of the proposed eigenfunction expansion. The aim is to improve convergence behavior of the eigenfunction expansions, especially in the case of formulations with significant convective effects, by simultaneously accounting for the relative importance of convective and diffusive effects within the eigenfunctions themselves, in comparison against the more traditional GITT solution path, which adopts a purely diffusive eigenvalue problem, and the convective effects are fully incorporated into the problem source term. After identifying a characteristic convective operator, and through a straightforward algebraic transformation of the original convection–diffusion problem, basically by redefining the coefficients associated with the transient and diffusive terms, the characteristic convective term is merged into a generalized diffusion operator with a space-variable diffusion coefficient. The generalized diffusion problem then naturally leads to the eigenvalue problem to be chosen in proposing the eigenfunction expansion for the linear situation, as well as for the appropriate linearized version in the case of a nonlinear application. The resulting eigenvalue problem with space variable coefficients is then solved through the GITT itself, yielding the corresponding algebraic eigenvalue problem, upon selection of a simple auxiliary eigenvalue problem of known analytical solution. The GITT is also employed in the solution of the generalized diffusion problem, and the resulting transformed ordinary differential equations (ODE) system is solved either analytically, for the linear case, or numerically, for the general nonlinear formulation. The developed methodology is illustrated for linear and nonlinear applications, both in one-dimensional (1D) and multidimensional formulations, as represented by test cases based on Burgers' equation.

Integral Transform Solutions for Diffusion in Heterogeneous Media

2008 ◽  
Author(s):  
Carolina P. Naveira ◽  
Olivier Fudym ◽  
Renato M. Cotta ◽  
Helcio R. B. Orlande
Keyword(s):  
Exact Solution ◽  
Eigenvalue Problem ◽  
Heterogeneous Media ◽  
Space Variable ◽  
Integral Transform ◽  
Heat Diffusion ◽  
Heat Transfer Analysis ◽  
Layered System ◽  
Varying Coefficients ◽  
Classical Integral

The Generalized Integral Transform Technique is employed in the hybrid numerical-analytical solution of heat diffusion problems in heterogeneous media. The GITT is utilized to handle the associated eigenvalue problem with aribitrarily space variable coefficients, defining an eigenfunction expansion in terms of a Sturm-Liouville problem of known solution. The formal solution is first applied in solving an example of space variable thermophysical properties found in heat transfer analysis of functionally graded materials (FGM), validated by the exact solution obtained through classical integral transforms in the specific situation of exponentially varying coefficients. Then, it is challenged in handling a double-layered system with abrupt variation of properties, and critically compared against the exact solution obtained by the classical integral transform method with the adequate discontinuous multi-region eigenvalue problem. The convergence behavior of the proposed expansions is then critically inspected and numerical results are presented to demonstrate the applicability of the general approach.

Unified Integral Transform Approach in the Hybrid Solution of Multidimensional Nonlinear Convection-Diffusion Problems

2010 ◽  
Author(s):  
Renato M. Cotta ◽  
Joa˜o N. N. Quaresma ◽  
Leandro A. Sphaier ◽  
Carolina P. Naveira-Cotta
Keyword(s):  
Open Source ◽  
Integral Transform ◽  
Nonlinear Partial Differential Equations ◽  
Integral Transforms ◽  
Analytical Methodology ◽  
Convection Diffusion ◽  
Development Platform ◽  
Unit Code ◽  
Nonlinear Transient ◽  
Diffusion Problems

The present work summarizes the theory and describes the algorithm related to the construction of an open source mixed symbolic-numerical computational code named UNIT — Unified Integral Transforms, that provides a development platform for finding solutions of linear and nonlinear partial differential equations via integral transforms. The reported research was performed by making use of the symbolic computational system Mathematica v.7.0 and the hybrid numerical-analytical methodology Generalized Integral Transform Technique — GITT. The aim here is to illustrate the robust and precision controlled simulation of multidimensional nonlinear transient convection-diffusion problems, while providing a brief introduction of this open source code. Test cases are selected based on nonlinear multi-dimensional formulations of the Burgers equations, with the establishment of reference results for specific numerical values of the governing parameters. Special aspects and computational behaviors of the algorithm are then discussed, demonstrating the implemented possibilities within the present version of the UNIT code.

scholarly journals Nonlinear eigenvalue problem in the integral transforms solution of convection-diffusion with nonlinear boundary conditions

2016 ◽  
Vol 26 (3/4) ◽  
pp. 767-789 ◽  
Author(s):  
Renato M Cotta ◽  
Carolina Palma Naveira-Cotta ◽  
Diego C. Knupp
Keyword(s):  
Boundary Condition ◽  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Condition ◽  
Nonlinear Eigenvalue Problem ◽  
Nonlinear Boundary Conditions ◽  
Content Type ◽  
Convection Diffusion ◽  
Diffusion Problems

Purpose – The purpose of this paper is to propose the generalized integral transform technique (GITT) to the solution of convection-diffusion problems with nonlinear boundary conditions by employing the corresponding nonlinear eigenvalue problem in the construction of the expansion basis. Design/methodology/approach – The original nonlinear boundary condition coefficients in the problem formulation are all incorporated into the adopted eigenvalue problem, which may be itself integral transformed through a representative linear auxiliary problem, yielding a nonlinear algebraic eigenvalue problem for the associated eigenvalues and eigenvectors, to be solved along with the transformed ordinary differential system. The nonlinear eigenvalues computation may also be accomplished by rewriting the corresponding transcendental equation as an ordinary differential system for the eigenvalues, which is then simultaneously solved with the transformed potentials. Findings – An application on one-dimensional transient diffusion with nonlinear boundary condition coefficients is selected for illustrating some important computational aspects and the convergence behavior of the proposed eigenfunction expansions. For comparison purposes, an alternative solution with a linear eigenvalue problem basis is also presented and implemented. Originality/value – This novel approach can be further extended to various classes of nonlinear convection-diffusion problems, either already solved by the GITT with a linear coefficients basis, or new challenging applications with more involved nonlinearities.

scholarly journals SOLUTION OF MULTIPHASE HEAT CONDUCTION PROBLEMS VIA THE GENERALIZED INTEGRAL TRANSFORM TECHNIQUE WITH DOMAIN CHARACTERIZATION THROUGH THE INDICATOR FUNCTION

2016 ◽  
Vol 15 (2) ◽  
pp. 53
Author(s):  
H. A. Machado ◽  
N. G. C. Leite ◽  
E. Nogueira ◽  
H. Korzenowisk
Keyword(s):  
Heat Conduction ◽  
Integral Transform ◽  
Heat Conduction Problem ◽  
Transport Equations ◽  
Indicator Function ◽  
Vast Number ◽  
Convection Diffusion ◽  
The Common ◽  
Integral Transform Technique ◽  
Diffusion Problems

The Generalized Integral Transform Technique (GITT) has appeared in the literature as an alternative to conventional discrete numerical methods for partial differential equations in heat transfer and fluid flow. This method permits the automatic control of the error and is easy to program, since there is no need for a discretization. The method has being constantly improved, but there still a vast number of practical problems that has not being solved satisfactory. In several brands of engineering, the transport equations have to be solved for a combination of different phases or materials or inside irregular domains. In this case, the mathematical resource of the Indicator Function can be employed. This function is a representation of the phases or parts of the domain with the numbers 0 and 1 for each phase. According to the method, the Indicator Function is defined by Poisson’s equation, which is added to the system of the transport equations. An integral is done along the curve that defines the interface that will generate the source term in Poisson’ equation used to calculate the Indicator Function distribution. The solution of the system of equations is done using the common GITT approach. Then, an analytical expression for each transformed potential of the indicator function and the other variables are available. Once the transformed potentials are known, the Indicator Function can be analytically operated, and the interface can be represented by an analytical continuous function. In this work, the use of the GITT in conjunction with the Indicator Function is proposed. The methodology is described and some previous results are presented. GITT is applied to a two-dimensional heat conduction problem in a multiphase domain with an irregular geometry, inside a square domain. The methodology presented here can be extended to all brands of convection-diffusion problems already solved via GITT.

scholarly journals Preconditioning and Uniform Convergence for Convection-Diffusion Problems Discretized on Shishkin-Type Meshes

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
Thái Anh Nhan ◽  
Relja Vulanović
Keyword(s):  
Uniform Convergence ◽  
Condition Number ◽  
Discrete System ◽  
Perturbation Parameter ◽  
Diffusion Problem ◽  
One Dimensional ◽  
Convection Diffusion ◽  
Proof Techniques ◽  
Consistency Error ◽  
Diffusion Problems

A one-dimensional linear convection-diffusion problem with a perturbation parameter ɛ multiplying the highest derivative is considered. The problem is solved numerically by using the standard upwind scheme on special layer-adapted meshes. It is proved that the numerical solution is ɛ-uniform accurate in the maximum norm. This is done by a new proof technique in which the discrete system is preconditioned in order to enable the use of the principle where “ɛ-uniform stability plus ɛ-uniform consistency implies ɛ-uniform convergence.” Without preconditioning, this principle cannot be applied to convection-diffusion problems because the consistency error is not uniform in ɛ. At the same time, the condition number of the discrete system becomes independent of ɛ due to the same preconditioner; otherwise, the condition number of the discrete system before preconditioning increases when ɛ tends to 0. We obtained such results in an earlier paper, but only for the standard Shishkin mesh. In a nontrivial generalization, we show here that the same proof techniques can be applied to the whole class of Shishkin-type meshes.

Vector eigenfunction expansion in the integral transform solution of transient natural convection

2019 ◽  
Vol 29 (8) ◽  
pp. 2684-2708 ◽  
Author(s):  
Kleber Marques Lisboa ◽  
Jian Su ◽  
Renato M. Cotta
Keyword(s):  
Natural Convection ◽  
Eigenvalue Problem ◽  
Eigenfunction Expansion ◽  
Stokes Equations ◽  
Integral Transform ◽  
Temperature Fields ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Content Type ◽  
Transient Natural Convection

Purpose The purpose of this work is to revisit the integral transform solution of transient natural convection in differentially heated cavities considering a novel vector eigenfunction expansion for handling the Navier-Stokes equations on the primitive variables formulation. Design/methodology/approach The proposed expansion base automatically satisfies the continuity equation and, upon integral transformation, eliminates the pressure field and reduces the momentum conservation equations to a single set of ordinary differential equations for the transformed time-variable potentials. The resulting eigenvalue problem for the velocity field expansion is readily solved by the integral transform method itself, while a traditional Sturm–Liouville base is chosen for expanding the temperature field. The coupled transformed initial value problem is numerically solved with a well-established solver based on a backward differentiation scheme. Findings A thorough convergence analysis is undertaken, in terms of truncation orders of the expansions for the vector eigenfunction and for the velocity and temperature fields. Finally, numerical results for selected quantities are critically compared to available benchmarks in both steady and transient states, and the overall physical behavior of the transient solution is examined for further verification. Originality/value A novel vector eigenfunction expansion is proposed for the integral transform solution of the Navier–Stokes equations in transient regime. The new physically inspired eigenvalue problem with the associated integmaral transformation fully shares the advantages of the previously obtained integral transform solutions based on the streamfunction-only formulation of the Navier–Stokes equations, while offering a direct and formal extension to three-dimensional flows.

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