A comparison of finite element and finite difference solutions of the one- and two-dimensional Burgers' equations

Abstract

Solutions to the one- and two-dimensional Burgers' equations with moderate to severe internal and boundary gradients have been used to compare minimum truncation error three, five-, and seven-point finite difference schemes with linear, quadratic, and cubic rectangular finite element schemes. The various schemes demonstrate the theoretically predicted convergence rates if the mesh is sufficiently refined. In one dimension the quadratic finite element scheme and the five-point finite difference scheme are computationally the most efficient. In multidimensions the finite element method is less economical than the finite difference method even if the group representation is used for the convective terms. In two dimensions the linear group finite element representation and the three-point finite difference scheme are computationally the most efficient on a coarse mesh or if a severe gradient is present.