The discontinuous Galerkin (DG) method has been widely adopted due to its excellent properties such as high accuracy and ease of parallelization. The adaptive mesh refinement (AMR) technique has been widely adopted to improve computational efficiency with much less computational cost compared with uniform global refinement to the same level with AMR. This paper combines the advantages of DG and AMR, and a new hybrid limiter is applied to the DG method on adaptive Cartesian grid based on p4est, an open-source library. The limiter exhibits advantages of high precision, compactness, robustness, and ease of implementation. The shock wave is captured with a shock indictor and the performance of the new hybrid limiter is compared with that of the total variational bounded (TVB) limiter in this paper. The result shows that the performance of the former is significantly better than that of the latter. A series of numerical examples for Euler equations and Navier-Stokes equations are used to verify the feasibility and efficiency of the proposed method. The results show that the new hybrid limiter performs very well in the AMRDG method, it has lower dissipation and great shock capture ability, and the computational efficiency is greatly improved while the accuracy is guaranteed.

DG  /  adaptive Cartesian grid  /  new hybrid limiter  /  shock capturing