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An adaptive gradient smoothing method (GSM) for fluid dynamics problems

In this paper, a novel adaptive gradient smoothing method (GSM) based on irregular cells and strong form of governing equations for fluid dynamics problems with arbitrary geometrical boundaries is presented. The spatial derivatives at a location of interest are consistently approximated by integrally averaging of gradients over a smoothing domain constructed around the location. Such a favorable GSM scheme corresponds to a compact stencil with positive coefficients of influence on regular cells. The error equidistribution strategy is adopted in the solution‐based adaptive GSM procedure, and adaptive grids are attained with the remeshing techniques and the advancing front method. In this paper, the adaptive GSM has been tested for solutions to both Poisson and Euler equations. The sensitivity of the GSM to the irregularity of the grid is examined in the solutions to the Poisson equation. We also investigate the effects of error indicators based on the first derivatives and second derivatives of density, respectively, to the solutions to the shock flow over the NACA0012 airfoil. The adaptive GSM effectively yields much more accurate results than the non‐adaptive GSM solver. The whole adaptive process is very stable and no spurious behaviors are observed in all testing cases. The cosmetic techniques for improving grid quality can effectively boost the accuracy of GSM solutions. It is also found that the adaptive GSM procedure using the second derivatives of density to estimate the error indicators can automatically and accurately resolve all key features occurring in the flow with discontinuities. Copyright © 2009 John Wiley & Sons, Ltd.