Variational Barrier Method of Adaptive Grid Generation in Hyperbolic Problems of Gas Dynamics

Abstract

Application of the harmonic mapping using a variational approach to generate moving adaptive grids in the hyperbolic problems of gas dynamics is considered. Using the example of a three-point model of adaptation, the possibility to generate an unfolded mesh with strong grid lines condensing in the vicinity of discontinuities of the control/(monitor) function is demonstrated. The algorithm of redistributing the boundary nodes is suggested and consists of using constrained minimization of the discrete harmonic functional when constraints define the boundary of the domain. In real computations due to mesh adaptation it is possible to reduce the errors, caused by shock waves smearing over the cells, by many factors of ten. Modeling of the two-dimensional (2-D) supersonic gas flow in the channel has shown that the same accuracy on the adaptive grid with the same structure as the quasi-uniform mesh can be acheived while requiring less CPU memory by a factor of 25 and less running time by a factor of 50 to 60. Computational tests of the steady transonic and supersonic flow over an airfoil demonstrate the ability of the method to control mesh sizes across shocks.