Spectral Difference Method for Unstructured Grids II: Extension to the Euler Equations

Abstract

An efficient, high-order, conservative method named the spectral difference method has been developed recently for conservation laws on unstructured grids. It combines the best features of structured and unstructured grid methods to achieve high-computational efficiency and geometric flexibility; it utilizes the concept of discontinuous and high-order local representations to achieve conservation and high accuracy; and it is based on the finite-difference formulation for simplicity. The method is easy to implement since it does not involve surface or volume integrals. Universal reconstructions are obtained by distributing solution and flux points in a geometrically similar manner for simplex cells. In this paper, the method is further extended to nonlinear systems of conservation laws, the Euler equations. Accuracy studies are performed to numerically verify the order of accuracy. In order to capture both smooth feature and discontinuities, monotonicity limiters are implemented, and tested for several problems in one and two dimensions. The method is more efficient than the discontinuous Galerkin and spectral volume methods for unstructured grids.

Large Eddy Simulation of Compressible Turbulent Channel Flow with Spectral Difference method

“…on complex geometries, has been the driving force for the development of higher order schemes for unstructured meshes such as the Discontinuous Galerkin (DG) Method, 1 Spectral Volume (SV) method 12,25 and Spectral Difference (SD) Method. 11,24 The SD method is a newly developed efficient high-order approach based on differential form of the governing equation. It was originally proposed by Liu et al 11 and developed for wave equations in their paper on triangular grids.…”

“…It was originally proposed by Liu et al 11 and developed for wave equations in their paper on triangular grids. Wang et al 24 extended it to 2D Euler equations on triangular grids and Sun et al 22 further developed it for three-dimensional Navier-Stokes equations on hexahedral unstructured meshes. The SD method combines elements from finite-volume and finite-difference techniques.…”

A Proof of the Stability of the Spectral Difference Method for All Orders of Accuracy

“…The SD method seems considerably simpler to implement than the DG method, and it has been observed by May [8] that it may be regarded as a variant of a pre-integrated nodal DG method. While the SD method has proved robust and productive in a variety of applications [9][10][11][12][13][14][15], doubts have been raised about its stability. In particular it has been suggested that the SD scheme is not stable for higher order triangular elements [16], and sometimes weakly unstable in one dimension depending on the choice of flux collocation points.…”

A p-Multigrid Spectral Difference Method For Viscous Compressible Flow Using 2D Quadrilateral Meshes

“…However, the need for highly accurate methods in applications such as large eddy simulation, direct numerical simulation, computational aeroacoustics etc., has seen the development of higher order schemes for unstructured meshes such as the Discontinuous Galerkin (DG) Method, 1,7 Spectral Volume (SV) method 9,10 and Spectral Difference (SD) Method. 4,5 The SD method is a newly developed efficient high-order approach based on differential form of the governing equation. It was originally proposed by Liu et al 4 and developed for wave equations in their paper on triangular grids.…”

“…It was originally proposed by Liu et al 4 and developed for wave equations in their paper on triangular grids. Wang et al 5 (2007) extended it to 2D Euler equations on triangular grids and Sun et al 3 (2007) further developed it for three-dimensional Navier-Stokes equations on hexahedral unstructured meshes. The SD method combines elements from finite-volume and finite-difference techniques, and is particularly attractive because it is conservative, and has a simple formulation and implementation.…”