Semi-Discrete Formulations for 1D Burgers Equation

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: R11 and R22 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 L2 and L∞ norm. We found that the R22 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 R11 Padé approximants. We note that the R22 Padé approximants softened the oscillations of the numerical solutions associated to the MEFG and SUPG formulations.