On the conservative and accurate CFD approximations for moving meshes and moving boundaries

We are concerned with the conservative approximations of compressible flows in the context of moving mesh or interface tracking. This paper makes a review of some numerical approaches, based on the Geometrical Conservation Law (GCL) property, for computation with moving mesh or moving boundary. Those methods are reformulated and some comparisons are made. We also propose a methodology to approximate accurately the mesh point trajectories and obtain accurate geometrical parameters (in 2D and 3D) satisfying the GCL property. For time accurate schemes, higher-order geometrical parameters also act as a stabilization of the approximation. The numerical effects of the higher-order trajectories are illustrated for a 2D moving airfoil. The efficiency of a mesh relaxation, based on the modified linear elasticity equations, is demonstrated for 3D computations of flow behavior in the combustion chamber of a four valves piston engine.