Implicit High-Order Spectral Finite Volume Method for Inviscid Compressible Flows

Abstract

The present work has the objective of demonstrating the capabilities of a spectral finite volume scheme implemented in a cell-centered finite volume context for unstructured meshes. The two-dimensional Euler equations are considered to represent the flows of interest. The spatial discretization scheme is developed to achieve high resolution for flow problems governed by hyperbolic conservation laws. Roe's flux difference splitting method is used as the numerical approximate Riemann solver. Several applications are performed in order to assess the method capability compared to data available in the literature and also compared to an weighted essentially nonoscillatory scheme. There is good agreement with the comparison data, and efficiency improvements over the weighted essentially nonoscillatory method are observed. The features of the present methodology include an implicit timemarching algorithm; second-, third-, and fourth-order spatial resolution; exact high-order domain boundary representation; and a hierarchical moment limiter to treat flow solution discontinuities.

A Detailed Comparison of Weno and SFV High-Order Methods for Inviscid Aerodynamic Applications

“…The CFD tools developed in the group and used in the present work have been verified and validated previously [2,3,4,5]. Furthermore, detailed analyses of the actual order of accuracy for the various schemes can also be found in the cited references.…”

“…State-of-the-art methods for unstructured, discontinuous highorder schemes include the Essentially Non-Oscillatory (ENO) and Weighted Essentially NonOscillatory (WENO) methods, Discontinuous Galerkin (DG), Spectral Finite Volume (SFV), Spectral Difference (SD) and Correction Procedure via Reconstruction (CPR) methods. In recent years, several efforts have been made by the CFD group of Instituto de Aeronautica e Espaco (DCTA/IAE) for the development of high-order unstructured grid schemes, such as the ENO, WENO and SFV methods, for the solution of aerodynamic flows with shock waves and contact surfaces [1,2]. Solution discontinuities pose a major challenge for research on highorder methods since reconstruction requires special treatment to preserve high-order accuracy and high resolution.…”

a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations

“…More adapted WENO schemes then have been developed [52,23,21] to overcome the problem of convergence toward the steady-state solution. We also mention some interesting developments for the steady-state case based on the Discontinuous Galerkin method [6,7,38], the Spectral Volume technique [10,18], and the Residual Distribution schemes [12,2,22].…”

High-Order Spectral Methods for Compressible Flows on Unstructured Meshes

“…The present work represents an effort to evaluate the SD method as an alternate high-order method for the applications of interest to the authors, namely external compressible aerodynamic flows. The CFD tools developed by the authors and used in the present work have been verified and validated previously [2,3,4].…”

“…The interested reader is referred to Ref. [3] for a throughout discussion of this subject. The cited reference also presents the curved boundary formulation.…”