A discontinuous and nonlinear multiscale method for solving convection- dominated problems

Resumo

This work presents a discontinuous and nonlinear multiscale method for solving problems dominated by convection. The method introduces a nonlinear artificial diffusion term at both scales of discretization while employing a discontinuous framework solely at the coarse scale. The micro scale is approximated using bubble functions, enabling efficient and accurate representation of the solution behavior. This approach aims to improve the accuracy and stability of numerical simulations for convection-dominated phenomena. Convection-dominated problems pose challenges in accurately resolving steep gradients and rapid variations in the solution. Conventional numerical methods often encounter problems related to numerical stability when solving this type of problem. In order to overcome these limitations, the proposed numerical scheme combines the benefits of nonlinear artificial diffusion, the discontinuous framework, and the use of bubble functions. To validate the effectiveness of the new method, some numerical experiments were conducted on convection-dominated problems of varying complexity. The results demonstrate that the multiscale method outperforms traditional approaches in accurately capturing the solution behavior, particularly in regions with sharp gradients.