Some Problems Multi-media Center Type ALE Method Hydrodynamics

In this thesis, we study some problems in computational fluid dynamics. It concerns three aspects:the first is the entropy fixed cell-centered Lagrangian scheme; the second is high-order moving mesh(abbr. HMM) kinetic scheme, finally we use the HMM scheme to simulate the chemical reaction fluids. More precisely, it includes the following three parts:Based on the work [P.-H. Maire, R. Abgrall, J. Breil, J. Ovadia, SIAM J. Sci. Comput.(2007)], we present an entropy fixed cell-contered Lagrangian scheme for solving the Euler equations of compressible gas dynamics. The scheme uses the fully Lagrangian form of the gas dynamics equations, in which the primary variables are cell-centered. And using the nodal solver, we obtain the nodal viscous-velocity, viscous-pressures, anti-dissipation velocity and anti-dissipation pressures of each node. The final nodal velocity is computed as a weighted sum of viscous-velocity and anti-dissipation velocity, so do nodal pressures, while these weights are calculated through the total entropy conservation for isentropic flows. Consequently, the constructed scheme is conservative in mass, momentum and energy, and preserves entropy for isentropic flows and satisfies a local entropy in-equality for non-isentropic flows. One-and two-dimensional numerical examples are presented to demonstrate theoretical analysis and performance of the scheme in terms of accuracy and robustness.We present a high-order moving mesh(abbr. HMM) kinetic scheme for com-pressible flow computations on structured and unstructured meshes. To construct the scheme, we employ the frame of the remapping-free ALE-type kinetic method [G.X. Ni, S. Jiang and K. Xu, J. Comput. Phys.(2009)] to get the discretization of compressible system. For the space accuracy, we use the WENO reconstruc-tion on the adaptive moving mesh from [H.Z. Tang, T. Tang, SIAM J. Numer. Anal.(2003)] to achieve time accuracy, for the time accuracy, we make use of the kinetic flux [K. Xu, J. Comput. Phys.(2001)] which includes time accu-rate integral, and thus obtain a HMM scheme. A number of numerical examples are given, especially an isentropic vortex problem to show the convergence order of the scheme. Numerical results demonstrate the accuracy and robustness of the scheme. For the multi material fluids simulation, coupling the Lagrangian method, which tracks material interfaces, with a remapping-free ALE-type ki-netic method within each single material region, a moving mesh BGK scheme for multi-material flow computations is proposed. Numerical examples shows that the current scheme keeps the interfaces sharp.Finally we consider the HMM scheme for reaction fluid, without splitting the system into a hydrodynamical part and an ODE part. We use the HMM scheme to calculating the system of gas dynamics equations coupled with the chemical reaction equation, which many different numerical schemes have simulated. The numerical algorithm is based on the GRP’s scheme [M. Ben-Artzi, J.Q. Li, G. Warnecke. J. Comput. Phys.(2006)] to increase the scheme-order in time and space. The variational approach is applied to generate the moving adaptive mesh. The numerical examples relating to the ZND detonation and unstable overdriven detonation are considered in one space dimensions.