Artificial boundary layers in discontinuous Galerkin solutions to shallow water equations in channels

Abstract

In this work, we consider the application of Discontinuous Galerkin (DG) solutions to open channel flow problems, governed by two-dimensional shallow water equations (SWE), with solid curved wall boundaries on which the no-normal flow boundary conditions are prescribed. A commonly used approach consists of straightforwardly imposing the no-normal flow condition on the linear approximation of curved walls. Numerical solutions indicate clearly that this approach could lead to unfavorable results and that a proper treatment of the no-normal flow condition on curved walls is crucial for an accurate DG solution to the SWE. In the test case used, errors introduced through the commonly used approach result in artificial boundary layers of one-grid-size thickness in the velocity field and a corresponding over-prediction of the surface elevation in the upstream direction. These significant inaccuracies, which render the coarse mesh solution unreliable, appear in all DG schemes employed including those using linear, quadratic, and cubic DG polynomials. The issue can be alleviated by either using an approach accounting for errors introduced by the geometric approximation or an approach that accurately represents the geometry.