Hybrid Finite Difference-Finite Volume Schemes on Non-uniform Grid

Abstract

High aspect ratio, skewness, non-orthogonality, and non-uniformity of grids have been a major issue for mesh developers for a long time. In this present work, we have taken an initial step for a numerical scheme that can handle any kind of mesh. Problems associated with non-uniformity of the grid (that can be related to aspect ratio in higher dimension) and achieving high order accuracy in those grids are discussed. A Hybrid Finite Difference-Finite Volume Method (Hybrid FD-FVM) which can retain high order accuracy on an arbitrary mesh by combining the advantage of higher order convergence property of finite difference method (FDM) and conservativeness property of FVM is presented. Though higher order version of FVM is available, they work well only on uniform meshes or slightly perturbed unstructured grid or gradually stretched grid. Smooth variation in meshes is recommended for CFD packages to obtain good accuracy—the reasons are discussed in the paper. FVM on the arbitrary mesh is at most second order accurate and are generally about first order accurate. Our method ensures higher order convergence on arbitrary non-uniform non-overlapped mesh but does not ensure complete numerical conservativeness on the non-uniform mesh for problems without shocks. The present work does not use volume averaging which is commonly used in FVM. The volume averaging works well in uniform mesh and can severely affect the result on the arbitrary varying non-uniform mesh. That also discussed here.