High-order-accurate fluctuation splitting schemes for unsteady hyperbolic problems using Lagrangian elements

Fluctuation splitting (FS) schemes have been developed in order to solve complex flows, using unstructured grids, more accurately than standard schemes, while employing a compact stencil. FS schemes, which can be viewed as an efficient implementation of finite elements within a residual distribution framework, have been applied with success to solve steady and unsteady, continuous and discontinuous compressible flows, see, e.g., [Bon05, Dep05]. In particular, for the case of unsteady advection problems, the authors derived the conditions to obtain consistent mass matrices [Dep05] and the sufficient conditions for an FS scheme to be (r+1)-th-order accurate in both space and time [Ros07]. Most importantly, they have proven that the explicit FS Lax-Wendroff scheme [Str94], being one of the FS scheme satisfying the above conditions, is an extremely efficient scheme with second-order accuracy in both space and time on a general triangulation composed by linear elements [Ros08]; indeed a remarkable result. This paper proceeds from the aforementioned studies to provide an analysis of FS schemes based on Lagrangian triangular elements, which allow one to achieve higher-order accuracy, while retaining the advantage of the compactness of the original schemes designed for linear elements. In particular, as done for the case of linear elements in [Dep05, Ros07], for the present case of triangular Lagrangian ones, the consistency conditions for the mass matrix are derived, together with those to be satisfied in order for mass lumping to preserve the accuracy of the scheme.