Upwind methods for the 1-D Euler equations, such as TVD schemes based on Roc's approximate Riemann solver, are reinterpreted as residual distribution schemes, assuming continuous piecewise linear space variation of the unknowns defined at the cell vertices. From this analysis three distinct steps are identified, each allowing for a multidimensional generalization without reference to dimensional splitting or I-D Riemann problems. A key element is the necessity to have continuous piecewise linear variation of the unknowns, requiring linear triangles in two space dimensions and tetrahedra in three space dimensions. Flux differences naturally generalize to flux contour integrals over the triangles. Rce's flux difference splitter naturally generalizes to a multidimensional flux balance splitter if one assumes that the parameter vector variable is the primary dependent unknown having linear variation in space. Nonlinear positive and second-order scalar distribution schemes provide a true generalization of the TVD schemes in one space dimension. Although refinements and improvements are still possible for all these elements, computational examples show that these generalizations present a new framework for solving the multidimensional Euler equations.

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A class of truly multidimensional upwind schemes for the computation of inviscid compressible flows is presented here, applicable to unstructured cell-vertex grids. These methods use very compact stencils and produce sharp resolution of discontinuities with no overshoots.

Schemes for twedimensional advection, based on the full advection direction, are derived and tested. The optimal, positive, linear scheme for triangles is presented and discussed. A technique for developing nonlinear schemes for linear problems is put forward, and positive, nonlinear schemes for triangles and quadrilaterals are presented. The linear schemes are based only on the advection direction and the mesh geometry; the nonlinear schemes add solutiongradient information to attain increased accuracy. All of the schemes are compact; the updates can be done on a cell-wise basis, using only the nodes that define that cell. All show a very marked improvement over mesh-aligned first-order upwind differencing, which employs the same stencil.

The multidimensional upwind approach for the Euler equations discussed in this paper generalizes to 2D the wellknown flux difference scheme of Roe. The method, which uses grids composed of triangles, is based on a conservative decomposition of the flux balance for each cell into scalar wave contributions, which are then upwinded to the vertices using a high-resolution compact monotone scalar advection scheme. Whereas the advection part and the conservative linearization have been extensively treated in the past, the present paper concentrates on the choice of the wave model generalizing the characteristic decomposition at the base of the 1D flux difference splitters. Contrary to the 1D case, many possibilities exist and a thorough review and comparison of existing and new models are given, emphasizing performance in subsonic as well as supersonic flows. The numerical results presented for a wide range of internal and external flows show the strong potential of the method.

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