Economically high‐order unstructured‐grid methods: Clarification and efficient FSR schemes

Abstract

In this article, we clarify reconstruction-based discretization schemes for unstructured grids and discuss their economically high-order versions, which can achieve high-order accuracy under certain conditions at little extra cost. The clarification leads to one of the most economical approaches: the flux-and-solution-reconstruction (FSR) approach, where highly economical schemes can be constructed based on an extended -scheme combined with economical flux reconstruction formulas, achieving up to fifth-order accuracy (sixth-order with zero dissipation) when a grid is regular. Various economical FSR schemes are presented and their formal orders of accuracy are verified by numerical experiments.

Implicit gradients based conservative numerical scheme for compressible flows

“…Also, such linearly high-order schemes have been demonstrated to serve as very low-dissipation schemes for practical turbulent-flow simulations [10,11,12,13]. As mentioned earlier, the developed schemes can be easily made to preserve high-order accuracy for nonlinear problems by applying economical high-order flux reconstruction techniques [18]. In the following sections, we will demonstrate that the developed schemes do indeed serve as very low-dissipation/dispersion schemes for strongly nonlinear problems with shock waves.…”

“…See Ref. [18] for efficient flux reconstruction techniques based on solution derivatives (not using flux derivatives). Alternatively, one may reinterpret the numerical solution values as cell averages and apply a single-point high-order flux quadrature formula proposed in Refs.…”

“…[16,17], methods based on the finitevolume method with point-valued solutions (instead of cell-averaged solutions) can achieve third-or higher-order accuracy for linear equations but can only be second-order accurate for nonlinear equations. These schemes can be made genuinely high-order accurate for nonlinear equations by introducing high-order flux reconstruction [18]. However, it has been demonstrated that linearly high-order schemes significantly improve the resolution of complex three-dimensional turbulent-flow simulations at a little additional cost over a baseline second-order scheme, even though not genuinely high-order accurate for such nonlinear problems [10,11,12,13].…”

Corrected Linear-Galerkin Schemes to Preserve Second-Order Accuracy for Cell-Centered Unstructured Finite Volume Methods

“…which was named "point-valued differential form with an approximate flux integral" in [64], and the only difference between the flux divergence and its average lies in an O h 2 truncation error [65],…”