The cell-centered discontinuous Galerkin method for Lagrangian compressible Euler equations in two-dimensions

• The system is discretized by the DG method. The obtain Lagrangian scheme is simple in form.
• Our method avoids computing the Jacobian matrix associated to the Lagrange–Euler flow map.
• The scheme is conservative for the mass, momentum and total energy.
• The scheme obeys the geometrical conservation law.
• The scheme maintains high-order accuracy.

This paper presents a new cell-centered Lagrangian scheme for two-dimensional compressible flow. The new scheme uses a semi-Lagrangian form of the Euler equations. The system of equations is discretized by discontinuous Galerkin (DG) method using the Taylor basis. The vertex velocities and the numerical fluxes through the cell interfaces are computed consistently by a nodal solver. The mesh moves with the fluid flow. The time marching is implemented by a class of the Runge–Kutta (RK) methods. A WENO reconstruction is used as a limiter for the RKDG method. The scheme is conservative for the mass, momentum and total energy, and obeys the geometrical conservation law. The scheme maintains high-order accuracy and has free parameters. Results of some numerical tests are presented to demonstrate the accuracy and the robustness of the scheme.