A multidimensional generalization of Roe's flux difference splitter for the euler equations

Abstract

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.

Time‐dependent algorithms for viscoelastic flow: Finite element/volume schemes

“…It is important to note that fluctuation distribution schemes have been developed within the realm of pure advection equations. For more details on the theory and the construction of such upwinding schemes, the reader is referred to the work of Jameson et al [18], Morton et al [19], Deconinck et al [15] and Hubbard et al [20].…”

“…To summarize, a time consistent distributioned approach over flux and source terms (area-weighted [15,16]), is paramount to ensure high-order accuracy at steady-state, and a consistent Median-Dual-Cell contribution is required to retain stability of convergence in complex flows. A generalised complete nodal update equation corresponds to the sum of contributions from all triangles surrounding node l, given by:…”

Constraint preserving schemes using potential-based fluxes. III. Genuinely multi-dimensional schemes for MHD equations

“…Considerable effort has been devoted for devising numerical methods which address the multi-dimensional character of nonlinear system such as (1.1). These methods include dimensional splitting [25], wave propagation algorithms [24,25], method of transport [15,16,34], bi-characteristics based evolution Galerkin methods [27,28] and fluctuation splitting schemes [12].…”

A multidimensional fluctuation splitting scheme for the three dimensional Euler equations

“…The extension of the fluctuation splitting method to non linear conservation laws is largely discussed in [9] and the conclusion is that the linearization L Z [6,20] has to be used as far as we have to get a constant approximation of whatever quantity.…”