Semi-Discrete Formulations for 1D Burgers Equation

TL;DR

This paper compares semi-discrete numerical methods for solving the 1D Burgers equation, highlighting the improved accuracy and convergence of R_{22} Padé approximants combined with finite element methods.

Contribution

It introduces the use of R_{22} Padé approximants with finite element methods, enhancing convergence and reducing oscillations in numerical solutions of the 1D Burgers equation.

Findings

R_{22} Padé approximants increase convergence region.

R_{22} approximants improve accuracy over R_{11}.

R_{22} approximants reduce oscillations in solutions.

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} Pad\'e 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} Pad\'e 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} Pad\'e approximants. We note that the R_{22} Pad\'e approximants softened the oscillations of the numerical solutions associated to the MEFG and SUPG formulations.