A higher order finite volume discretization based on Staggered Update Procedure (SUP)

Abstract

The unstructured data-based finite volume solvers are widely adopted in industrial computations owing to their robustness and flexibility in handling complex geometries. Most of these solvers use linear reconstruction methods that compute convective fluxes to second-order accuracy, but viscous fluxes typically achieve first order accuracy on irregular meshes. This leads to reduced overall solution accuracy and impacts the convergence rate. With the advancement of high-performance computing (HPC) resources and growing demand of high-fidelity CFD simulations such as LES and DES, there is a pressing need for more accurate numerical methods.

This thesis introduces a higher order accurate, robust, and computationally efficient finite volume formulation based on a Staggered Update Procedure (SUP) for solving compressible viscous flows on unstructured meshes. The essential idea in SUP lies in updating the solution not only at cell centres, as in traditional finite volume methods, but also at mesh vertices. The vertex values are computed using an Upwind-Least Squares Finite Difference (LSFD-U) approach. A key innovation is the development of the defect correction technique integrated with the Green-Gauss theorem based gradient computation, which recovers second order accuracy even on highly skewed meshes where the traditional Green-Gauss based gradients become inconsistent. This ensures that the viscous fluxes are computed to the same order of accuracy as that of the convective fluxes when used with a linear reconstruction step.

The SUP methodology is rigorously tested on linear convection-diffusion problems in two- and three-dimensions, as well as on benchmark laminar flow cases such as flow past NACA 0012 airfoil and around circular cylinder. Results show significantly improved solution accuracy, reduced numerical diffusion, and enhanced convergence behaviour compared to traditional finite volume method. To address the computational cost associated with the recursive defect correction process, the gradient computation is offloaded to GPU using OpenACC. This GPU-accelerated implementation achieves performance gains of 9× in 2D and 28× in 3D, effectively masking the additional computational effort while fully leveraging modern parallel hardware.