An Accuracy Assessment of Cartesian-Mesh Abstract Approaches for the Euler Equations William J. Coirier* NASA Lewis Research Center Cleveland, Ohio USA and Kenneth G. Powell** The University of Michigan Ann Arbor, Michigan USA I Introduction A critical assessment of the accuracy of Cartesian-mesh approaches for solving the Euler equations is made. An exact solution of the Euler equations (Ringleb'sflow) is used not only to infer the order of error of the Cartesian- mesh approaches,but also to compare the magnitude of the error directly to that obtained with a structured mesh approach. The effect of cell merging is investigated as well as the use of two different K-exact reconstruction proce- dures. The solution methodology of the schemes is explained and tabulated results are presented to compare the solution accuracies.Adaptive and uniform mesh refinement is evaluatedfor Ringleb'sjlow and the super- sonic flow through an d-symmetric inlet. * Member AIAA. Aerospace Engineer, DoctDral Candi- date, The University of Michigan, Department of Aero- space Engineering ** Senior Member AIAA. Associate Professor, Depart- ment of Aerospace Engineering Copyright 1993 by the American Institute of Aeronautics and Astranau- tics, Inc. No copyright is asserted in the United States under Etle 17, U.S. Code. I h e U.S. Govennnent has a myalty-free license to exercise all rights under the copyright claimed herein for Governmental purposes. All other rights are reserved by the copyright owner. With the advent of unstructured meshes, it is becoming pos- sible to perform high quality calculations of flows of increas- ing geometric and physical complexity. This geometric flexibility is obtained by using an unstructured grid data structurethat, if formulated properly, can allow mesh enrich- ment by cell division. In this way, unstructured solvers with adaptive mesh refinementcan resolve disparate length scales on geometrically complicated domains and perhaps provide a means to achieve automatic mesh convergence. Mesh redistribution schemes have the benefit of being able to use existing, structured mesh flow solvers with few modifica- tions, but suffer from the constraints borne by the structured mesh data structure. As pointed out in [I], adaptive mesh refinement via mesh or cell enrichment is superior to mesh redistribution for precisely this reason, although both schemes can be an improvement to the non-adaptive approach. Through the proper formulation of data structures and by an efficient implementation of non-traditional algo- rithms, unstructured meshes approaches can compete with and complement standard, structured-mesh approaches. The method assessed here is a Cartesian-mesh approach. Cartesian-mesh approaches have been in the literature for a number of years. In [21 and [4] unsteady shock hydrody- namic problems were computed on a Cartesian mesh on a Cartesian domain. Adaptive mesh refinement was achieved by adding collections of cells, grouped into contiguous grids about fronts in the field, using front detection algorithms