A simple and efficient space-time adaptive grid technique for unsteady compressible flows

A space-time adaptive gridding technique for unsteady flows is presented. The technique is applied to the 2D unsteady Euler equations. The method is relatively simple, computationally very efficient and it can be easily adapted to other types of fluid flow. It consists of four parts: (i) a time-stepping algorithm that adapts the grid to the solution several times per coarse time step, (ii) a second-order accurate discretisation of the flow equations that combines a limited upwind discretisation of the fluxes with a two-step discretisation of the time derivatives and is well-suited for adapted grids, (iii) a simple data structure to store the solution and the grid geometry, and (iv) a refinement criterion. Two of these are tested, one based on the first and one on the second spatial derivative of the density. Results for two test problems, the classical forward-facing step problem and the shedding of vorticity from a flat plate, show that the method is much more efficient than comparable methods without adaptive gridding.