THE GOAL OF THIS PROPOSAL IS TO DEVELOP AN EFFICIENT AND HIGH-ORDER-ACCURATE METHOD FOR SIMULATING HYPERSONIC FLOWS WITH SHOCKS (I.E. A REENTRY VEHICLE). THE BASIC IDEA IS TO COMBINE AN ARBITRARY-LAGRANGIAN-EULERIAN (ALE) MESH MOVEMENT SIMULATION THAT TRACKS THE SHOCKS WITH A HIGH-ORDER ADAPTIVE FINITE ELEMENT METHOD. THE ADVANTAGES OF THIS APPROACH OVER SHOCK CAPTURING WOULD BE IMMENSE. WHEREAS SHOCK CAPTURING METHODS ARE LIMITED TO FIRST ORDER ACCURACY WE PLAN TO DEMONSTRATE THAT WITH A MOVING MESH ALE METHOD ANY ORDER OF ACCURACY CAN BE OBTAINED BY INCREASING THE POLYNOMIAL DEGREE OF THE FINITE ELEMENT BASIS. FURTHERMORE THE SHOCK JUMP WILL NOT BE SPREAD OVER THE GRID BUT WILL BE MAINTAINED AS A DISCONTINUITY THAT EXACTLY SATISFIES THE RANKINE-HUGONIOT EQUATIONS WITH NO OVERSHOOTS OR UNDERSHOOTS. NOT ONLY WILL THIS MAKE THE SIMULATIONS MORE RELIABLE IT WILL ALLOW ORDERS-OF-MAGNITUDE IMPROVEMENTS IN ACCURACY GIVEN THE SAME COMPUTATIONAL RESOURCES