While solving numerically a problem initially formulated on an unbounded domain, one typically truncates this domain, which necessitates developing a special closing procedure at the newly formed external boundary. The most efficient and most popular approach to obtaining such closures is through setting the so-called artificial boundary conditions (ABCs). The issue of ABCs appears most significant in many areas of scientific computing, for example, in CFD, where external problems represent a wide class of practically important applications and where the proper treatment of external boundaries may have a profound impact on the overall performance of numerical algorithms and interpretation of the results.
Most of the currently used techniques for setting the ABC's can basically be classified into two groups. The methods from the first group (global ABCs) usually provide for high accuracy and robustness of the numerical procedure but often appear to be relatively cumbersome and expensive. The methods from the second group (local ABCs) are, as a rule, algorithmically simple, numerically cheap, and geometrically universal; however, they usually lack accuracy of computations. Dr. S. Tsynkov had developed and implemented a new ABCs technique that combines the advantages relevant to the methods of both types. The technique is based on application of the difference potentials method of V. S. Ryaben'kii. This approach enables one to obtain highly accurate ABCs in the form of certain boundary operator equations. The operators involved are analogous to the pseudodifferential boundary projections introduced by A. P. Calderon.
Numerical experiments for different flow regimes, subsonic (including very low Mach numbers) and transonic, laminar and turbulent, in both two and three space dimensions, including three-dimensional flows with propulsive jets, show that the new ABCs clearly outperform the standard methods from the standpoints of accuracy, multigrid convergence rate, and robustness.
Here we provide a brief summary of computational results for a transonic turbulent flow around the ONERA M6 wing. These results clearly demonstrate the speedup of multigrid convergence due to implementations of the new boundary conditions.