An Adaptive Projection Method for the Incompressible Euler Equations

In this paper we present a method for solving the time-dependent incompressible Euler equations on an adaptive grid. The method is based on a projection formulation in which we first solve convection equations to predict intermediate velocities, and then project these velocities onto a space of approximately divergence-free vector fields. Our treatment of the convection step uses a specialized second-order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid flow. Our approach to adaptive refinement uses a nested hierarchy of grids with simultaneous refinement ofthe grids in both space and time. The integration algorithm on the grid hierarchy is a recursive procedure in which a coarse grid is advanced, fine grids are advanced multiple steps to reach the same time as the *This work of these authors was performed under the aus-· pices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under contract W-7405-Eng-4S. Support under contract W-7405-Eng-48 was provided by the AMS and HPCC Programs of the DOE Office of Scientific Computing and by the Defense Nuclear Agency under IACRO 93-S17. tResearch supported at UC Berkeley by DARPA and th<; National Science Foundation under grant DMS-S919074; by a National Science Foundation Presidential Young Investigator award under grant ACS-S9fi8522; and by the Department of Energy High Performance Computing and Communications Program under grant DE-FG03-92ER25140. 1 coarse grid and the grids are then synchronized. We will describe the integration algorithm in detail, with emphasis on the projection used to enforce the incompressibility constraint. Numerical examples are presented to demonstrate the convergence properties of the method and to illustrate the behavior of the method at the interface between coarse and fine grids. An additional example demonstrates the performance of the method on a more realistic problem.