A space and time fixed point mesh adaptation method

We introduce a space-time metric-based approach for the best set of spatial meshes $M_{opt}\left(t\right)$ combined with the best set time step ${\tau }_{opt}\left(t\right)$ of space-time complexity $N_{\text{st}}$ for the calculation of transient flows with implicit time advancing. Both types of error estimate, feature-based and goal-oriented, are considered for the compressible RANS equations. The case of a mesh which is adapted at each time step, and the case where the mesh is constant during a time subinterval of the whole simulation are theoretically analyzed. These space-time error estimates are then considered inside a Global Transient Fixed Point mesh adaptation algorithm. Applications to flow with vortex shedding past a cylinder are then described.