Recent decades, the Discontinuous Galerkin method (DGM) has gained a resurgence of interest by its high-order accuracy over the conventional second-order accurate finite volume method (FVM) with combination of both advantages of FVM and Finite Element Method (FEM). In respect of DGM, the high-order accuracy is obtained by increasing the order of approximate polynomials rather than relying on extended stencils in classical FVM. Besides, the numerical flux at the interface is developed by arising from solutions of the Riemann problems in the case of FVM. Other than that, DGM maintains the compactness because only data from neighbors of element boundaries is required, which results in that the inter-element communication is minimal and boundary conditions are implemented in a straightforward way. Also, due to the locality of the DGM, refining or coursing the meshes is easily handled without the restrictions on the continuity with neighbor elements. In addition, DGM is suited for complex geometries based on the unstructured meshes. As a result, it owns the attractive features of being easily extended to high-order approximation, well suited for complex geometries, highly parallelizable and easily handling adaptive strategies.
The DGM was originally introduced to solve the neutron transport problems. Then it is developed by Cockburn and Shu for solving the nonlinear systems of hyperbolic conservation problems. They presented the Runge-Kutta DG method, which used DGM for spatial discretization and the TVD Runge-Kutta method for time integration. However, the convergence rate is extremely slow for large-scale simulations because of poor CFL stability condition. To solve this problem, an implicit time integration method is adopted. In the current research, a high-order implicit DGM flow solver for the three-dimensional problems has been developed on tetrahedral grids. A fully implicit method based on Euler backward differencing and linearization of the residuals is adopted to effectively obtain the steady state solutions. Several numerical simulations have been tested to estimate the accuracy of DGM simulation results and the efficiency of obtaining the steady solutions.
A three-dimensional high-order flow solver based on a DGM has been developed for the numerical simulations on unstructured meshes. The numerical experiments have been tested to evaluate the efficiency and accuracy. The subsonic flow past a bump in a channel and a sphere has been simulated by the DGM solver. In addition, a subsonic flow past a fixed NACA 0015 wing was also numerically simulated for the demonstration of the efficiency and high-order of accuracy. Specifically, in the case of NACA 0015 wing, DGM does a better job at preserving the wing tip vortex strength comparing to other high-order numerical schemes. Then, the flow around a Caradonna-Tung rotor in hover is simulated to validate the flow solver by means of comparing the computed pressure coefficients with experimental data. All the numerical results have demonstrated that the implicit DGM with high-order representation of curved solid boundaries can achieve the designed high-order of accuracy and efficiency in obtaining steady state solutions.
