A robust high-order residual distribution type scheme for steady Euler equations on unstructured grids

A robust high-order algorithm is proposed to solve steady Euler equations on unstructured grids. The main ingredients of the algorithm include a standard Newton method as the outer iterative scheme and a linear multigrid method as the inner iterative scheme with the block lower-upper symmetric Gauss–Seidel (LU-SGS) iteration as its smoother. The Jacobian matrix of the Newton-iteration is regularized by the local residual, instead of using the commonly adopted time-stepping relaxation technique based on the local CFL number. The local Jacobian matrix of the numerical fluxes are computed using the numerical differentiation, which can significantly simplify the implementations by comparing with the manually derived approximate derivatives. The approximate polynomial of solution on each cell is reconstructed by using the values on centroid of the cell, and limited by the WENO hierarchical limiting strategy proposed by Xu et al. [Z.L. Xu, Y.J. Liu, C.W. Shu, Hierarchical reconstruction for discontinuous galerkin methods on unstructured grids with a WENO-type linear reconstruction and partial neighboring cells, Journal of Computational Physics 228 (2009) 2194–2212]. It is found that the proposed algorithm is insensitive to the parameters used. More precisely, in our computations, only one set of the parameters (namely, the proportional constant α for the local residual, the relaxation parameter τ in the Newton-iteration, the weight μ in the WENO scheme and the number of smoothing steps in the multigrid solver) is employed for various geometrical configurations and free-stream configurations. The high-order and robustness of our algorithm are illustrated by considering two-dimensional airfoil problems with different geometrical configurations and different free-stream configurations.