A High-Order Discontinuous Galerkin Method for Solving Preconditioned Euler Equations

Abstract

A high-order discontinuous Galerkin (DG) method is presented for solving the preconditioned Euler equations with an explicit or implicit time marching scheme. A detailed description is given of a practical implementation of a precondition matrix of the type of Weiss and Smith and of the DG spatial discretization scheme employed, with particular emphasis on the artificial viscosity-based shock capturing techniques. The curved boundary treatment is proposed through adopting a NURBS surface equipped with a radial basis function interpolation to propagate the boundary displacement to the interior of the mesh. The resulting methods are verified by simulating flows over two-dimensional airfoils, such as symmetric NACA0012 or asymmetric RAE2822, and over three-dimensional bodies, such as an academic hemispherical headform or aerodynamic ONERA M6 wing. Numerical results show that the present method functions for both transonic and nearly incompressible flow simulations, and the proposed treatment of curved boundaries, play an important role in improving the accuracy of the obtained solutions, which are in good agreement with available experimental data or other numerical solutions reported in literature.

Keywords

• discontinuous Galerkin
• Euler equations
• preconditioning
• shock capturing
• curved boundary treatment