In this paper, we construct a high-order conservative cell-centered Lagrangian type discontinuous Galerkin (DG) scheme for the two-dimensional compressible Euler equations on straight-edge meshes. This high-order scheme is a highly non-trivial extension of the second-order pure Lagrangian DG scheme in our earlier work [15]. Unlike the previous second order pure Lagrangian methods, the basic idea of this high order scheme is that only nodes move in a pure Lagrangian fashion with the fluid velocity, while points connecting two nodes as a straight-line edge are subject to the flux exchange to maintain high order accuracy. On the cell edges containing the material interfaces and contact discontinuities, the flux term of this DG scheme is designed following that of our previous second-order pure Lagrangian scheme to strictly ensure that there is no mass exchange across these edges. In this way, the Lagrangian type DG scheme on the straight-edge meshes can not only achieve arbitrarily high order accuracy in smooth regions away from material interfaces and contact discontinuities, but also capture the material interfaces or contact discontinuities automatically and sharply. This high-order Lagrangian type DG scheme avoids curvilinear meshes, thus it is simple to implement and can save computational cost effectively. Another contribution of this work is that the oscillation-free (OF) procedure is introduced for the first time into the Lagrangian type method. In the Lagrangian framework, the shape of the meshes constantly changes and could be extremely non-uniform, which causes difficulties in controlling numerical oscillations for high order approximations. In this work, the OF procedure does not depend on any problem-related parameters, can effectively suppress oscillation near discontinuities without compromising the original high-order accuracy and conservation properties, and avoids additional time step size constraint caused by the numerical damping. Finally, a series of numerical experiments are presented to demonstrate the good performance of our high-order Lagrangian type DG scheme in terms of accuracy, resolution for discontinuities, and non-oscillation.
