International Journal of Computational Fluid Dynamics, 2013 Vol. 27, Nos. 4–5, 210–227, http://dx.doi.org/10.1080/10618562.2013.813491 A mass-matrix formulation of unsteady fluctuation splitting schemes consistent with Roe’s parameter vector Aldo Bonfiglioli a,∗ and Renato Paciorri b a Scuola di Ingegneria, Universit` a della Basilicata, Viale dell’Ateneo Lucano 10, Potenza, Italy; b Dipartimento di Meccanica e Aeronautica, Universit` a di Roma, La Sapienza, Via Eudossiana, Roma, Italy (Received 20 December 2012; final version received 5 June 2013) A mass-matrix formulation of the fluctuation splitting schemes for solving compressible, unsteady flows is proposed. This formulation is consistent with the conservative linearisation based on parameter vector and allows to extend to unsteady flows the ‘invariance under similarity transformations’ property that had been shown to hold for the steady version of the schemes. Second-order time accuracy is achieved using a Petrov–Galerkin finite element interpretation of the fluctuation splitting schemes. The approach may however be readily applicable to all other time-accurate fluctuation splitting formulations that have been so far proposed in the literature. Applications of the proposed methodology to two- and three-dimensional, inviscid and viscous compressible flows are reported and discussed in the paper. Keywords: fluctuation splitting; residual distribution; compressible flows; unsteady flows; mass matrix; low Mach number; linearisation; parameter vector 1. Introduction Almost the entirety of the CFD codes used for solving the compressible Euler and Navier–Stokes equations rely upon a locally one-dimensional (1D) Riemann problem in the direction normal to the control volume (CV) boundary as the key ingredient for modelling wave propagation phe- nomena, even in multi-dimensional flows. For this kind of solver, the order of accuracy of the spatial discretisation is determined by the order of the polynomial which is used to reconstruct the set of dependent variables on either side of the interface separating adjacent CVs. Linear reconstruc- tion, which is the de facto standard in most state-of-the-art research and commercial codes, leads to second order of accuracy in space. Higher spatial accuracy can be achieved by employing a higher (than linear) functional representa- tion of the dependent variables within the CVs. Different approaches have emerged over the last years; these include: k-exact Finite Volume (FV), Spectral Volume (SV) and Dis- continuous Galerkin (DG) Finite Element (FE) methods, to name just a few. The paper by Wang (2007) provides a com- prehensive review of high-order methods on unstructured grids. Following a different route, fluctuation splitting (FS) schemes abandon the locally 1D model based on the solu- tion of the Riemann problem and take a multi-dimensional approach to wave propagation. Introduced in the late 1980s by Roe (1987), FS schemes employ a vertex-centred stor- age of the unknown and piece-wise linear representation of ∗ Corresponding author. Email: aldo.bonfiglioli@unibas.it the dependent variables, which vary continuously through the cell interfaces. FS schemes that are second-order ac- curate both in space and time are nowadays sufficiently well understood; a specific aspect of second-order-accurate unsteady FS schemes forms the subject of the present paper. Higher (than second) order accurate FS schemes have also been proposed in the literature, see e.g. Abgrall and Roe (2003b), Hubbard and Laird (2005), Ricchiuto et al. (2008) and Rossiello et al. (2010), and are currently being ac- tively developed, but they will not be covered in the present contribution. One distinctive feature of second-order-accurate FS schemes, when compared to their FV counterpart having the same spatial order of accuracy, is the compactness of the stencil, which is limited to the set of nearest (or distance- 1) neighbours. In FV schemes, on the contrary, even the use of the simple linear reconstruction leads to a com- putational stencil that encloses the distance-2 neighbours. Despite the use of a more compact stencil, second-order- accurate FS schemes are reported, see e.g. Bastin and Rog´ e (1999), Wood and Kleb (1999) and Guzik and Groth (2008), to achieve lower discretisation error levels than FV ones. Moreover, the compactness of the stencil has obvious bene- ficial consequences in terms of implementation of boundary conditions, parallelisation efficiency and increased matrix sparsity when implicit time integration is used. Another distinctive feature of FS scheme is that, at least for a perfect gas, a multi-dimensional version (Deconinck, C  2013 Taylor & Francis