Implicit Gradient Reconstruction for Unstructured Mesh Finite Volume Method

Abstract

In the present work, an Implicit Gradient Reconstruction (IGR) method is proposed in the context of Finite Volume Methodology (FVM). There are three computationally intensive steps involved in a typical finite volume framework for a spatially second-order accurate upwind scheme. These are solution reconstruction, solution limiting and flux finding. Solution re- construction involves determination of gradients while solution limiting requires comparison of double precision numbers. Further, computation of gradient and solution limiting is cell based procedure while flux finding is edge/face based procedure. The proposed IGR procedure, which is edge/face based procedure, integrates all these steps obviating explicit reconstruction and limiting steps resulting in a considerable reduction in computational effort and associated memory footprint. In the modified CIR (MCIR) scheme of linear convection equation, a parameter φ is in- troduced to control dissipation. The IGR procedure is derived from the MCIR scheme. The relation between the φ parameter and solution reconstruction in a finite volume procedure is systematically established. The methodology is extended to multidimensions, where the use of φ implicitly represents a reconstruction step. Hence, this procedure is referred to as Implicit Gradient Reconstruction. In addition, it is brought out that the use of φ also serves the purpose of solution limiting. The spatial accuracy of this procedure is demonstrated by computing the 2-D circular convection problem. The methodology, when extended to the Euler equations of Gas dynamics, results in the reconstruction of the characteristic variables. Consequently, three steps in the computation of explicit residual, namely, solution reconstruction, limiting and flux computation, are seamlessly merged into a single step. Owing to its significantly smaller memory footprint, the procedure is particularly relevant to large scale parallel computing. This procedure can be effortlessly incorporated into any of the existing finite volume solvers where inviscid flux formulation is based on characteristic decomposition. The capability of IGR procedure is established through several test cases involving inviscid and viscous flow computations in one and two dimensions. The results obtained from IGR procedure are compared with reconstruction based solvers and wind tunnel data wherever available. The IGR procedure produces results comparable to the classical reconstruction based procedure. The aforementioned IGR procedure is applicable to any flux formulation involving charac- teristic decomposition. An Edge Based Reconstruction Limiting (EBRL) is proposed for flux formulation not involving characteristic decomposition. This procedure is very simple and does not require classical diamond path reconstruction. The results obtained from EBRL based Roe and AUSM Plus flux formulation on transonic viscous flow over RAE 2822 airfoil are very good and compare well with wind tunnel experiments.