Numerical methods for simulating subsonic flows and solving inverse problems based on the new two-dimensional Euler equations in Lagrangian formulation. A transformation to derive the Euler equations in Lagrangian plane to simplify the computational grid and to minimize the numerical diffusion caused by the convective flux. Constructed is a numerical scheme, the dimensional-splitting with hybrid upwind flux operators, where the two-dimensional Riemann problem was solved in the Lagrangian plane. The method, solving the inverse geometry shape design problem in Lagrangian plane, gives the flow filed results and the geometry shape concurrently.