New least squares method with geometric conservation law (GC-LSM) for compressible flow computation in meshless method

In this study, we propose a meshless scheme, GC-LSM (Geometric Conservation Least Squares Method), satisfying the geometric conservation and 1st order consistency. These constraints are introduced in order to overcome the non-conservativeness of the original meshless scheme and imposed by Lagrange multiplier on the least squares method which determines weighting coefficients of the derivative terms. Improvements on the meshless scheme are confirmed through computations with randomly distributed grid points for a sine wave, nozzle flow, and hypersonic flow around blunt body. Combined with AUSMPW + and MUSCL scheme, GC-LSM of the second order accuracy gives non-oscillating solution around a strong shockwave, even for hypersonic flow, and shows its capability comparable to the finite volume method in views of accuracy, robustness, and convergence.