An analysis of the stability of the least square finite difference scheme and its shock capturing ability in inviscid flows

Abstract

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.

A study of the role of weights in the spectral properties of the least square finite difference scheme and the least square upwind finite difference scheme

“…In Katz, 41 the selection of a proper domain of influence is done using weighting functions. To guarantee the reciprocity of nodes, weights are used in each local connectivity such that certain constraints are satisfied and in Mungur et al 16 the existence of an optimum weight guaranteeing such reciprocity is shown analytically. Oñate et al 42 and Oñate and Idelsohn 43 developed a method that incorporated different least squares weighting methods to improve the accuracy of derivatives and formulations for higher order methods.…”

“…If a > 0, D − i is chosen while when a < 0, D + i is chosen in Equation (5). In Mungur et al, 16 a detailed stability analysis is conducted for the first order LSFD scheme. A bounded l ∞ norm for the first order LSFD scheme is obtained using a conditional timestep given by 1 2…”

“…To solve the linear advection equation using the first order forward in time LSFD‐U given as If , is chosen while when , is chosen in Equation (5). In Mungur et al, 16 a detailed stability analysis is conducted for the first order LSFD scheme. A bounded norm for the first order LSFD scheme is obtained using a conditional timestep given by …”

“…Katz and Jameson 37 have developed a polynomial least square which uses weights to force alignment of the derivative approximation, thereby diagonalising the least square matrix and thus decreasing the computational time to resolve the solution. In Mungur et al, 16 it is proved that there exists an optimum weight that solve the optimization problem used to calculate weights.…”

“…Liu 15 gave a detailed discussion about a weak and strong meshless formulation with their associated advantage and disadvantage. An analysis of the stability of the least square finite difference (LSFD) scheme is conducted in Mungur et al 16 The scheme with added diffusion is analyzed and applied to inviscid flows. This scheme is then made second order using the concept of modified difference and shows the ability to resolve shock wave, rarefaction wave and contact discontinuity quite accurately.…”