The Discontinuous Galerkin Method For Solving Lagrangian Euler Equations And Second-order Conservative Remapping Algorithm

The compressible Euler equations have several different expressions in the La-grangian formalism. Based on the (semi-)Lagrangian differential form of the compress-ible Euler equations, we derive the weak form of them. This weak form is discretized by the discontinuous Galerkin (DG) method and two new cell-centered Lagrangian schemes are obtained. In the fist Lagrangian scheme, the flux is approximated by the Lax-Friedrichs (L-F) and the Harten-Laxvan Leer contact wave (HLLC) numerical flux and the Roe average algorithm [Cheng J, Shu CW, J Comput Phys,(2007)] are used to determine the vertex velocity. The time marching is implemented by a class of Runge-Kutta (RK) type methods corresponding to the same order of the spatial dis-cretization. In order to control spurious numerical oscillation near the discontinuity, a slope limiter is used in the scheme. The scheme is conservative for the mass, momen-tum and total energy, is essentially non-oscillatory, and can achieve uniformly second order accuracy on moving and distorted Lagrangian meshes. One and two-dimensional numerical examples are presented to demonstrate the good performance of this scheme.The above-mentioned Lagrangian scheme has many advantages, but it does not obey the geometrical conservation law. This may be relevant to the vertex velocity, the numerical flux, the limiter et al. So in order to improve the robustness of the above-mentioned scheme, the second new Lagrangian scheme is given. In this scheme the weak form is also discretized by the RKDG method. The differences are:this new scheme uses the nodal solver in [Maire PH, Abgrall R, Breil J, et al. SIAM J Sci Comput,(2007)] to determine the vertex velocity and the numerical flux, and the inner side velocity is equal to the average of the vertex velocity. In addition, we use the limiter in [Jia ZP, Zhang SD, J Comput Phys,(2011)] to control spurious numerical oscillation. The new scheme is not only conservative for the mass, momentum and total energy, but can also obey the geometrical conservation law. The scheme maintains high-order accuracy. Results of some numerical tests are presented to demonstrate the accuracy and the robustness of the scheme.When using Lagrangian scheme to solve the Euler equations, the very fact that computational cells exactly follow fluid particles can result in severe grid deformation, moreover causing inaccuracy and even breakdown of the computation. So it needs re-zoning meshes and remapping physical quantities when the computational cells deform severely. Based on the above Lagrangian schemes in this thesis, we give a conserva-tive remapping scheme which transfer the Lagrangian solution in the old grid onto the new grid. The remapping scheme has two steps:the first step is using the remapping method in [Margolin LG, Shashkov M, J Comput Phys,(2003)] to obtain the approx-imate mean values in the new cells, then using the repair algorithm in [Kucharik M, Shashkov M,Wendroff B, J Comput Phys,(2003)] to ensure the mean values in the range of local bounds; the second step is using the mean values to reconstruct the linear polynomial in the new cell, and using Van Leer limiter to limit the gradient of this linear polynomial to ensure no new extremum. The new remapping method is second-order accurate, conservative and bound-preserving.