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In this talk, we will present our recent results[1] on behaviors of numerical solutions to the strong shock interaction for moving mesh schemes based on the one-dimensional Godunov and HLLC Riemann solvers.When the grid motion velocity is close to Lagrangian one, these Godunov methods may suffer from numerical shock instability.In order to cure such instability, we construct a new cell centered ALE algorithm for inviscid, compressible gas flows.The main feature of the algorithm is to introduce a nodal contact velocity and ensure the compatibility between edge fluxes and the nodal flow intrinsically.Numerical experiments show that the scheme is robust on both quadrilateral and triangular grids and reduces numerical shock instability.