On the connection between the spectral volume and the spectral difference method

In the last few years, two new high-order accurate methods for unstructured grids have been developed, namely the spectral volume (SV) and the spectral difference (SD) method. Both methods are related to the well-known discontinuous Galerkin (DG) method, see for instance [1], in the sense that they also use piecewise continuous polynomials as solution approximation space. The development of the SV method was mainly reported in publications by Wang [2], Wang et al. [3–5,8], Liu et al. [6] and Sun et al. [7]. Further contributions were made by Chen [12,13], Van den Abeele et al. [14] and Van den Abeele and Lacor [15]. The development of the SD method was reported more recently in publications by Liu et al. [9] and Wang et al. [10]. In 1D, the SD method is identical to the multi-domain spectral method proposed by Kopriva [11]. The SV method is strongly related to the finite volume (FV) method. As with the FV method, the SV solution variables are averaged values over control volumes (CVs) and the residuals corresponding to these solution variables can be written as the sum of the fluxes through the CV faces. The SV method differs from the FV method in the choice of the stencils used to approximate the fluxes through these CV faces. By partitioning each cell, or spectral volume (SV), into CVs in a similar way, a unique stencil, valid for all cells in the mesh, can be obtained. This is the major advantage of the SV method over high-order accurate unstructured FV methods, where these stencils depend on the local mesh geometry. At faces between two SVs, a Riemann solver is used to compute a unique flux from the two solutions in the neighbouring SVs. Similarly, the SD method is strongly related to the finite difference (FD) method. For both the FD and the SD method, the solution variables are pointwise values, for which the residuals can be written as a function of the flux derivatives in the solution points. With the SD method the computational domain is subdivided in