Comparison of the ENATE approach and discontinuous Galerkin spectral element method in 1D nonlinear transport equations

In this paper a comparison of the performance of two ways of discretizing the nonlinear convection-diffusion equation in a one-dimensional (1D) domain is performed. The two approaches can be considered within the class of high-order methods. The first one is the discontinuous Galerkin method, which has been profusely used to solve general transport equations, either coupled as the Navier-Stokes equations, or on their own. On the other hand, the ENATE procedure (Enhanced Numerical Approximation of a Transport Equation), uses the exact solution to obtain an exact algebraic equation with integral coefficients that link nodal values with a three-point stencil. This paper is the first of a thorough assessment of ENATE by comparing it with well established high-order methods. Several test cases of the steady Burgers’ equation with and without source have been chosen for comparison.