One Type Of Adaptive Heat Conduction Viscosity To Ameliorate Wall Heating Errors In Lagrangian Coordinates

We study the wall heating problem of shock capturing schemes in the Lagrangian methods and give an explanation of its origins and an adaptive approach to ameliorate this phenomenon.Due to the anomalous phenomenon such as wall heating of shock capturing schemes in Lagrangian formalism are caused by viscosity and dissipation mechanisms of the modified equations of conservation laws, so decreasing the effect of the viscosity and diffusing the errors to the whole computing area are the two main methods we consider. Using as high order and high resolution methods as possible and adding heat flux are the two approaches to implement, the effect of the latter one is what we mainly investigate here. It is also necessary to find some conditions to distinguish the areas where wall heating problem exit, which is another main job in this paper, because we hope to implement our method only in this kind of areas to keep the resolution in the vicinity of contact discontinuity.We first give the High order and High resolution Godunov-type schemes of ID Euler equations in the Lagrangian formalism. We give an adaptive method to find the contact and shock discontinuity, and design a form of heat flux based on the energy flux provided by the HLL Riemann solver. Secondly, we extend our method to cell-centered Lagrangian schemes to control the wall heating problem for two dimensional compressible flow problems. For the specialty of the Riemann solvers in this kind of method which only involves the use of HLLC or sound wave approximation, the heat flux performs well in dealing with the wall heating problem. At the end of the paper, we study the Lagrangian method for one dimensional MHD problems. We give the HLLC-MHD in Lagrangian framework, which pave the foundation for the future work of the Lagrangian method of two dimensional MHD problems, and then extend our method of controlling the wall heating to computations of MHD problems.The results prove the efficiency and the satisfaction of our newly invented methods, which can improve the performance of the computations of the strong shocks.