New Gradient Calculation Method for MUSCL Type CFD Schemes in Arbitrary Polyhedra

The gradient calculation in the reconstruction phase is the key for the spatial accuracy and robustness of MUSCL type CFD schemes. For the gradient calculation, which bears central role in the second order reconstruction, new method for arbitrary polyhedra based on WLSQ (weighted least square) method merging the benefit of G-G (Green-Gauss) method is presented. The method, named GLSQ (G-G based WLSQ), has second order spatial accuracy for non-orthogonal and non-uniform linear mesh and gives rigorous gradient for thin curved mesh for which LSQ shows huge error. A new geometrical monotonicity condition, which gradient calculation method should satisfy for robustness, is also introduced. Although GLSQ is originally developed for hybrid meshes that combines octree and layered meshes, it is shown numerically that it also has better accuracy in usual rectangle and triangle meshes.