A novel efficient reconstruction scheme for unstructured grids based on iterative least-squares methods

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 first-order least-squares methods, and then converges to higher-order values through iteration. The scheme is defined only by the values stored in the face-adjacent cells. The size of the stencils and reconstruction matrix, 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 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 k -exact reconstruction with the same order of accuracy.