Integration strategy for high-order discontinuous Galerkin (DG) method for solving Euler equations

In the full paper, according to the DG method, we derive the accuracy requirements of numerical integration for line integral and volume integral in the Euler equations discretized by unstructured grid. Considering the above accuracy requirements, we summarize: (1) the minimal numbers of integral nodes used in the line integral with the Gauss-Legendre quadrature rule and the Gauss-Lobatto quadrature rule respectively, (2) the minimal number of integral nodes used in the volume integral with the Gauss quadrature rule, (3) a new quadrature method based on reconstruction. By using a numerical example, we evaluate: (1) the accuracy requirements for the numerical integration method, (2) the effects of different numerical integrations on computing stability, accuracy and efficiency. The numerical simulation results, given in Figs. 2 through 6, and their analysis show preliminarily that: (1) in case the accuracy requirement of line integral is not fulfilled, the stability and the convergence of DG method can not be ensured; the accuracy requirement of volume integral has strong influence on the accuracy of flow solution; (2) the Gauss-Legendre quadrature rule is more efficient than the Gauss-Lobatto quadrature rule for the same accuracy of numerical integration; (3) for high-order DG discretization, our quadrature method based on reconstruction is more accurate than the classical Gauss quadrature rule although its efficiency is lower by approximately 15%.