Semi-Discrete Formulations for 1D Burgers Equation

Abstract

In this work we compare semi-discrete formulations to obtain numerical solutions for the 1D Burgers equation. The formulations consist in the discretization of the time-domain via multi-stage methods of second and fourth order: R_{11} and R_{22} Padé approximants, and of the spatial-domain via finite element methods: least-squares (MEFMQ), Galerkin (MEFG) and Streamline-Upwind Petrov-Galerkin (SUPG). Knowing the analytical solutions of the 1D Burgues equation, for different initial and boundary conditions, analyzes were performed for numerical errors from L_{2} and L_{\infinity} norm. We found that the R_{22} Padé approximants, added to the MEFMQ, MEFG, and SUPG formulations, increased the region of convergence of the numerical solutions, and showed greater accuracy when compared to the solutions obtained by the R_{11} Padé approximants. We note that the R_{22} Padé approximants softened the oscillations of the numerical solutions associated to the MEFG and SUPG formulations.

14 pages, 2 figures, in Portuguese