High-order upwind method for hyperbolic grid generation

Abstract

A new hyperbolic grid generation method based on a high-order accurate upwind approximation is presented and developed for two- and three-dimensional spaces in this paper. The method uses a system of hyperbolic equations for grid generation. Due to a hyperbolic property of the grid generation equations, there exist characteristic directions and high-order TVD upwind scheme can be applied to the directions. The method employs a three-point backward difference scheme for descritization of a derivative in a marching direction, while a high-order accurate upwind scheme using MUSCL interpolation with minmod limiter is employed according to the plus or minus sign of eigenvalues of coefficient matrices to descritize derivatives in the other directions. Making use of the upwind scheme the present hyperbolic grid generation method needs no artificial damping term, which a conventional hyperbolic grid generation method requires to prevent oscillations. The method incorporates a subiteration procedure to solve nonlinear equations at each marching step in order to obtain both sufficient robustness and satisfactory grid orthogonality realized by the high-order upwind scheme. For two dimensions, the Newton subiteration is performed at each marching step before advancing to next grid line. For three dimensions, a new pseudo-time subiteration approach is introduced and incorporated in the method to achieve further robustness. It is demonstrated that in the hyperbolic grid generation the first-order accurate upwind scheme is too dissipative to generate an orthogonal grid, although the higher-order accurate upwind scheme can generate the orthogonal grid. The generated grids around bodies of complex geometry with sharp edges, deep concave and high convex show excellent orthogonality and smoothness, and demonstrate promising properties of the method.