A cell-centered Lagrangian discontinuous Galerkin method using WENO and HWENO limiter for compressible Euler equations in two dimensions

This article introduces a new variant of the cell-centered Lagrangian discontinuous Galerkin method for two-dimensional compressible flow which was proposed by Jia et al. (J Comput Phys 230(7):2496–2522, 2011). Unlike the original scheme of Jia, in this article, the Euler equations in the Lagrangian formalism are discretized in the reference frame using a bilinear map and a corresponding Jacobian matrix. The working variables are the conservative ones. The Taylor basis functions defined in the reference coordinates provide the piece-wise polynomial expansion of the variables. A limited linear polynomial DG scheme and an HWENO DG scheme are presented which use WENO and HWENO reconstruction algorithms as limiters, respectively. For both schemes, the vertex velocities and the numerical fluxes through the cell interfaces are computed consistently by virtue of a nodal solver. The time marching is implemented by a class of TVD Runge–Kutta type methods. The schemes are conservative for the mass, momentum, and total energy. They are implemented on quadrilateral linear meshes and can achieve second-order accuracy both in space and time. Results of some numerical tests are presented to demonstrate the accuracy and the robustness of the scheme.