An analysis of the stability of the least square finite difference scheme and its shock capturing ability in inviscid flows. (1st January 2019) - Mungur, Maheshsingh; Dauhoo, Muhammad Zaid; Elahee, Mohammad Khalil
Summary: A meshless method – The Least Square Finite Difference scheme (LSFD) with diffusion is analyzed and applied to inviscid flows. The scheme is made second‐order by using a modified difference in the formulation of LSFD. Several numerical experiments, namely the Sod shock tube and the shallow water problems, are carried out and, in the limelight of the results obtained, the ability of the scheme to resolve shock wave, rarefaction wave, and contact discontinuity is discussed. The conditional stability of the LSFD scheme is established. The LSFD uses weights to diagonalize the least square matrix resulting in the spatial discretization in order to gain computational time. We prove that there exists a unique weight for the resulting optimization problem. The weighted version of LSFD is used to solve the isentropic vortex problem numerically and the results are used to discuss the dissipative nature of the scheme. Five configurations of the two‐dimensional Riemann problems are used in our numerical experiments. The capability of the scheme to capture the complex interaction of multiple planar waves is discussed in the limelight of the results on the Riemann problems. The result of the shock reflection problem shows that the scheme is minimally dissipative and leads to sharp and well‐resolved shocks. Abstract : We provide an analysis of the Least Square Finite Difference (LSFD) scheme and propose a new second‐order accurate LSFD using the concept of modified difference. The robustness of this scheme is tested extensively on inviscid flow problems with complex flow patterns. Numerical experiments show that the second‐order scheme provides better numerical resolution and numerical studies are conducted to assess the accuracy. The figure shows the results of a 2D Riemann problem containing 2 forward and 2 backward shocks.