Unified Integral Transforms Algorithm for Solving Multidimensional Nonlinear Convection-Diffusion Problems

Abstract

The present work summarizes the theory and describes the algorithm related to an open-source mixed symbolic-numerical computational code named unified integral transforms (UNIT) that provides a computational environment for finding hybrid numerical-analytical solutions of linear and nonlinear partial differential systems via integral transforms. The reported research was performed by employing the well-established methodology known as the generalized integral transform technique (GITT), together with the symbolic and numerical computation tools provided by the Mathematica system. The main purpose of this study is to illustrate the robust precision-controlled simulation of multidimensional nonlinear transient convection-diffusion problems, while providing a brief introduction of this open source implementation. Test cases are selected based on nonlinear multidimensional formulations of Burgers’ equation, with the establishment of reference results for specific numerical values of the governing parameters. Special aspects in the computational behavior of the algorithm are then discussed, demonstrating the implemented possibilities within the present version of the UNIT code, including the proposition of a progressive filtering strategy and a combined criteria reordering scheme, not previously discussed in related works, both aimed at convergence acceleration of the eigenfunction expansions.

Acknowledgments

The authors would like to acknowledge the financial support provided by CNPq and FAPERJ, both in Brazil.

Notes

(*)UNIT code with Gaussian integration {12, 12, 12} points and squared eigenvalues reordering.

(**)UNIT code with Gaussian integration {12, 12, 12} points and combined reordering.

(⁺)UNIT code with user provided analytical integration.

^a Semi-analytical integration (N = 45 and M = {16, 16, 16}).

^b,d estimate for 3 terms residue.

^c Gaussian quadrature (N = 45 and M = {16,16,16}); and

^e UNIT with user provided analytical integration (N = 300).

^a Semi-analytical integration (N = 50 and M = {28,28}).

^b,d estimate for 3 terms residue.

^c Gaussian quadrature (N = 50 and M = {28,28}); and

^e Mathematica 7.

^a Semi-analytical integration (N = 35 and M = 180).

^b,d estimate for 3 terms residue.

^c Gaussian quadrature (N = 35 and M = 180); and

^e Mathematica 7.