# Article

**Summary:**

A space-time formulation for unsteady inviscid compressible flow computations in 2D moving geometries is presented. The
governing equations in Arbitrary Lagrangian-Eulerian formulation (ALE) are discretized on two layers of space-time finite elements connecting levels $n$, $n+1/2$ and $n+1$. The solution is approximated with linear variation in space (P1 triangle) combined with linear variation in time. The space-time residual from the lower layer of elements is distributed to the nodes at level $n+1/2$ with a limited variant of a positive first order scheme, ensuring
monotonicity and second order of accuracy in smooth flow under a time-step restriction for the timestep of the first layer. The space-time residual from the upper layer of the elements is distributed to both levels $n+1/2$ and $n+1$, with a similar scheme, giving monotonicity without any time-step restriction. The two-layer scheme allows a time marching procedure thanks to initial value condition imposed on the first layer of elements. The scheme is
positive and second order accurate in space and time for arbitrary meshes and it satisfies the Geometric Conservation Law condition (GCL) by construction. Example calculations are shown for the Euler equations of inviscid gas dynamics, including the 1D problem of gas compression under a moving piston and transonic flow around an oscillating NACA0012 airfoil.