Recent advances in computational-analytical integral transforms for convection-diffusion problems

Abstract

An unifying overview of the Generalized Integral Transform Technique (GITT) as a computational-analytical approach for solving convection-diffusion problems is presented. This work is aimed at bringing together some of the most recent developments on both accuracy and convergence improvements on this well-established hybrid numericalanalytical methodology for partial differential equations. S p e c i a l e m p h a s i s i s g i v e n t o n o v e l a l g o r i t h m implementations, all directly connected to enhancing the eigenfunction expansion basis, such as a single domain reformulation strategy for handling complex geometries, an integral balance scheme in dealing with multiscale problems, the adoption of convective eigenvalue problems in formulations with significant convection effects, and the direct integral transformation of nonlinear convection-diffusion problems based on nonlinear eigenvalue problems. Then, selected examples are presented that illustrate the improvement achieved in each class of extension, in terms of convergence acceleration and accuracy gain, which are related to conjugated heat transfer in complex or multiscale microchannel-substrate geometries, multidimensional Burgers equation model, and diffusive metal extraction through polymeric hollow fiber membranes. Numerical results are reported for each application and, where appropriate, critically compared against the traditional GITT scheme without convergence enhancement schemes and commercial or dedicated purely numerical approaches.

Analysis of conjugated heat transfer in micro-heat exchangers via integral transforms and non-intrusive optical techniques

“…Following the GITT formalism, it is desirable that the eigenvalue problem be chosen in order to contain as much information as possible about the original problem. The eigenvalue problem is formulated by directly applying separation of variables to problem (3) so that all the information concerning the transitions of the original subdomains are represented within the eigenvalue problem, by means of the space variable coefficients k(y,z) and w(y,z):…”

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

“…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].…”

“…Following the formalism in the GITT [26][27][28][29][30][31][32][33][34]71,72], an eigenvalue problem is chosen that provides the base for the eigenfunction expansion. Here, when Eqs.…”

“…It would not take long for the GITT to be challenged by the solution of fluid flow problems governed either by the boundary layer equations or the Navier-Stokes equations [24,25]. Since then, the hybrid method was progressively extended and new classes of problems and applications have been dealt with, and it has been reviewed at different stages and sources [26][27][28][29][30][31][32][33][34].…”

Improving the precision of discrete numerical solutions using the generalized integral transform technique

“…The GITT has been successfully employed to generate benchmark results for many problems in thermal and fluids science and engineering [5,6,11,13]. Recent applications of the technique include solution of conduction-radiation problems [24], convection-diffusion problems [9,10], conjugated heat transfer [2,14], fermentation kinetics [23], among others. The success of the GITT is due to its global error control capabilities; when the finite summation series achieves solution convergence in the flow region with steepest gradients in both time and space, according to any arbitrarily specified accuracy criteria, the solution will be at least as accurate everywhere else in the flow.…”