A granular computing method for nonlinear convection-diffusion equation

Abstract

This paper introduces a method of solving nonlinear convection-diffusion equation (NCDE), based on the combination of granular computing (GrC) and characteristics finite element method (CFEM). The key idea of the proposed method (denoted as GrC-CFEM) is to reconstruct the solution from coarse-grained layer to fine-grained layer. It first gets the nonlinear solution on the coarse-grained layer, and then the function (Taylor expansion) is applied to linearize the NCDE on the fine-grained layer. Switch to the fine-grained layer, the linear solution is directly derived from the nonlinear solution. The full nonlinear problem is solved only on the coarse-grained layer. Numerical experiments show that the GrC-CFEM can accelerate the convergence and improve the computational efficiency without sacrificing the accuracy.