The Unified Integral Transforms (Unit) Algorithm With Total and Partial Transformation

Cotta, Renato M.; Knupp, Diego C.; Naveira-Cotta, Carolina P.; Sphaier, Leandro A.; Quaresma, João Nazareno Nonato

doi:10.1615/computthermalscien.2014008663

Cited by 27 publications

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“…The Generalized Integral Transform Technique (GITT) [26][27][28][29][30][31][32][33][34], based on the classical integral transform method [10], provides a hybrid numerical-analytical nature to the eigenfunction expansion approach, yielding error-controlled solutions to a large number of linear and nonlinear convection-diffusion problems. The basic steps in the GITT algorithm can be summarized as follows [71,72] More recently, nonlinear eigenvalue problems have also been employed with marked improvement on convergence [33,74]; c) Develop the integral transform pair and obtain the transform and inversion, that will define the transformation operation and the explicit recovering of the potential; d) Solve the eigenvalue problem, either in analytical form and symbolic computation, or through the GITT approach itself, transforming the chosen differential eigenvalue problem into an algebraic one [23,26]. A convergence acceleration strategy, based on integral balances, has been recently advanced in handling eigenvalue problems through the GITT [75]; e) Integral transform the original PDE and obtain the transformed differential system, which shall be an ODE system for a total transformation, when all the independent variables are eliminated except one.…”

Section: The Generalized Integral Transform Techniquementioning

confidence: 99%

A review of hybrid integral transform solutions in fluid flow problems with heat or mass transfer and under Navier–Stokes equations formulation

Cotta

Lisbôa

Curi

et al. 2019

Numerical Heat Transfer, Part B: Fundamentals

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The Generalized Integral Transform Technique (GITT) is reviewed as a hybrid numericalanalytical approach for fluid flow problems, with or without heat and mass transfer, here with emphasis on the literature related to flow problems formulated through the full Navier-Stokes equations. A brief overview of the integral transform methodology is first provided for a general nonlinear convection-diffusion problem. Then, different alternatives of eigenfunction expansion strategies are discussed in the integral transformation of problems for which the fluid flow model is either based on the primitive variables or the streamfunction-only formulations, as applied to both steady and transient states. Representative test cases are selected to illustrate the different eigenfunction expansion approaches, with convergence being analyzed for each situation. In addition, fully converged integral transform results are critically compared to previously reported simulations obtained from traditional purely discrete methods.

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Section: The Generalized Integral Transform Techniquementioning

confidence: 99%

A review of hybrid integral transform solutions in fluid flow problems with heat or mass transfer and under Navier–Stokes equations formulation

Cotta

Lisbôa

Curi

et al. 2019

Numerical Heat Transfer, Part B: Fundamentals

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“…(20e), the nonlinear source term vanishes, and an exact solution of the transformed ODE system, Eqs. (25), is obtained, and the final solution is given explicitly as…”

Section: Test Casesmentioning

confidence: 99%

“…which has a straightforward analytical solution, while the transformed system would still have the form of Eqs. (25), but with the following transformed source term and initial condition: g i ðt;…”

Section: Test Casesmentioning

confidence: 99%

“…In problems with more significant convective effects, this solution path may lead to slower convergence rates in the eigenfunction expansions. Such difficulties have been overcome through the use of convergence acceleration schemes, such as with analytical filtering and integral balance procedures [11,21,22], or by adopting a partial integral transformation scheme, which transforms only those spatial variables in which the diffusive effects predominate, yielding a transformed partial differential system to be numerically solved [23][24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

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

Cotta

Naveira-Cotta

Knupp

2017

Journal of Thermal Science and Engineering Applications

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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.

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“…Deve ser ressaltado que neste trabalho optou-se por um esquema de transformação parcial, onde somente uma das coordenadas espaciais, onde o efeito difusivoé predominante,é transformada [4], resultando num sistema de equações diferenciais parciais unidimensional a ser resolvido numericamente.…”

Section: Problema De Autovalor E Problema Transformadounclassified

Simulação Computacional do Transporte Horizontal Bidimensional de Contaminantes no Rio Macaé Através da Técnica da Transformada Integral Generalizada

Souza¹,

Knupp²,

Rodrigues³

et al. 2015

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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do Estado do Rio de Janeiro, IPRJ/UERJ, Nova Friburgo, RJ Resumo. Neste trabalho foi desenvolvida uma solução híbrida analítico-numérica através da Técnica da Transformada Integral Generalizada (GITT) empregando-se recursos de programação simbólica através do Software Wolfram Mathematica 10, para solução de um problema bidimensional transiente de dispersão de contaminantes em corpos d'água. Para tal, foi realizada a simulação do comportamento do lançamento instantâneo de um contaminante conservativo (NaCl) em um trecho situado na parte inferior do rio Macaé, localizado no nordeste do Estado do Rio de Janeiro. Para melhor convergência da solução híbrida foram aplicados filtros analíticos para homogeneizar os contornos, e consequentemente, a equação transformada em um sitema acoplado de equações diferenciais parciais unidimensional, resolvido numericamente através de rotinas intrísecas do Mathematica. Os resultados foram comparados com dados experimentais disponíveis, e os resultados obtidos indicam a plausibilidade da solução deste problema através do GITT, o que pode levar a ganhos computacionais, importantes para aplicação em problemas inversos em trabalhos futuros.Palavras-chave. Simulação Ambiental, Contaminantes, GITT, Rio Macaé IntroduçãoDiversos problemas que envolvem fenômenos naturais, físicos, biológicos e sociais podem ser descritos matematicamente por equações diferenciais parciais. Contudo, em grande parte dos casos, a geometria do meio e as condições de contorno inviabilizam a possibilidade de aplicação de técnicas de solução analíticas ou até mesmo de métodos 1 alrsouza@iprj.uerj.br 2

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The Unified Integral Transforms (Unit) Algorithm With Total and Partial Transformation

Cited by 27 publications

References 0 publications

A review of hybrid integral transform solutions in fluid flow problems with heat or mass transfer and under Navier–Stokes equations formulation

A review of hybrid integral transform solutions in fluid flow problems with heat or mass transfer and under Navier–Stokes equations formulation

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

Simulação Computacional do Transporte Horizontal Bidimensional de Contaminantes no Rio Macaé Através da Técnica da Transformada Integral Generalizada

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