Desenvolvimento e simulação computacional de esquemas upwind baseados em curvas de Bézier e interpolação de Hermite

Abstract

This thesis presents new families of upwind schemes for approximating convective terms in partial differential equations within finite differences and finite volumes. The families of schemes termed SOBUS (Second Order Bézier Upwind Scheme), PUBICK (Piecewise Upwind Bézier Interpolation for Convective Kinematics) and CUBICK (Cubic Upwind Bézier Interpolation for Convective Kinematics) are proposed based on quadratic and cubic Bézier curves, respectively. In contrast, the FDHPUS family (Fifth-Degree Hermite Polynomial Upwind Scheme) is based on Hermite interpolation. The PUBICK, CUBICK, and FDHPUS schemes satisfy stability criteria and constraints to ensure a bounded solution, such as TVD (Total Variation Diminishing), CBC (Convection Boundedness Criterion), and BAIR (Boundedness, Accuracy, and Interpolative Reasonability). The convergence of the numerical solutions was analyzed when the PUBICK, CUBICK, and FDHPUS schemes were adopted to solve a series of 1D and 2D problems. The numerical results of these schemes were compared with those of other established schemes in the literature, showing good agreement with exact solutions and numerical data. The results of this thesis suggest that upwind schemes based on Bézier curves and Hermite interpolation are promising alternatives for solving convection-dominated problems.