New hybrid compact schemes for structured irregular meshes using Birkhoff polynomial basis

Abstract

This paper discusses the formulation of a class of hybrid compact schemes for structured irregular meshes. First, we derived three- and five-point compact schemes with required boundary closures for non-periodic problems. Second, we devised hybrid compact schemes from the explicit representation of the first- and second-order derivatives on irregular staggered and collocated meshes, respectively. For staggered meshes, hybrid interpolation schemes are also formulated. The main advantage of the hybrid formulation is that it requires boundary closures only for the compact scheme for first-order derivatives on the collocated mesh. Resolution and added numerical diffusion properties of hybrid schemes are assessed by using global spectral analysis (Sengupta et al., 2003). Direct numerical simulations of 2D dispersive shallow water and incompressible Navier-Stokes equations at different Reynolds numbers with fourth-order RK4 time integration method are considered to validate the accuracy and efficiency of new schemes. Numerical solutions are also compared with results reported in the literature.