A Novel Efficient Reconstruction Scheme for Unstructured Grids based on Iterative Least-Squares methods

Abstract

A new compact higher-order variable reconstruction scheme based on iterative least-squares methods is proposed. The approximation of derivatives is split into multi-step linear least-squares methods, and then converges to higher-order values through iteration. The scheme is defined only by the values stored in the faceadjacent cells. The size of the stencils and reconstruction matrix is small, and thus the computational cost and memory consumption is significantly reduced. In a time-evolutional problem, the converged value at the previous time-step is used as an initial value of the iteration of the reconstruction in order to achieve quick convergence. In addition, a WENO-like weight function is implemented for shock-capturing problems. In a vortex-advection problem, it is shown that only one iteration of the reconstruction per time-step gives sufficient convergence, and that higher-order accuracy is achieved efficiently. Then a double-Mach reflection problem is simulated. The present scheme shows high resolution of the unsteady flow structure, and no severe numerical instability is observed. The computational cost of the fourth-order iterative reconstruction is cheaper than the conventional -exact reconstruction with the same order of spatial accuracy.