Corrected Linear-Galerkin Schemes to Preserve Second-Order Accuracy for Cell-Centered Unstructured Finite Volume Methods (Advances in Applied Mathematics and Mechanics, 2024)
Unstructured finite volume methods are typically categorized based on control volumes into the node- and cell-centered types. Because of certain inherent geometric properties, the second-order Linear-Galerkin scheme, favored for its simplicity and ability to preserve second-order solution accuracy, is predominantly appliedto node-centered control volumes. However, when directly applied to cell-centeredcontrol volumes, the designed solution accuracy can be lost, particularly on irregulargrids. In this paper, the least-square based Linear-Galerkin discretization is extendedto arbitrary cell-centered elements to ensure the second-order accuracy can always beachieved. Altogether four formulations of corrected schemes, with one being fullyequivalent to a conventional second-order finite volume scheme, are proposed andexamined by problems governed by the linear convective, Euler and Navier-Stokesequations. The results demonstrate that the second-order accuracy lost by the originalLinear-Galerkin discretization can be recovered by corrected schemes on perturbedgrids. In addition, shock waves and discontinuities can also be well captured by corrected schemes with the help of gradient limiter function.