Residual Distribution Schemes: Foundations and Analysis

Abstract

In this article, we introduce and analyze the class of numerical schemes known as residual distribution or fluctuation splitting schemes. The root of this family of discretizations is found mainly in works aiming at generalizing the finite volume techniques to a genuinely multidimensional upwinding context, as in, for example (Hall, Morton, Ni, Roe). However, the final result is a numerical method sharing a lot with other techniques, such as finite element schemes (Carette, Deconinck, Paillere and Roe, 1995). Our aim is to look at fluctuation splitting/residual distribution methods from a generic theoretical point of view. We discuss in a general way the basic principles and the criteria used in the design of the schemes: positivity, k‐exacteness, energy stability, well‐balancedness. We also show similarities with known, more classical, discretization techniques, as well as the main distinguishing features that make residual distribution methods a valid alternative to these classical schemes. Finally, we present some numerical applications and comparisons. Throughout the text, we give an extensive overview of the existing literature, and finally conclude the chapter by reviewing most of the ongoing research on the topic.

An ALE Formulation for Explicit Runge-Kutta Residual Distribution

“…Of course, the main step is the decomposition (24). We need to design the sub-residuals φ K j in such a way that stability and convergence is garantied.…”

“…We need to design the sub-residuals φ K j in such a way that stability and convergence is garantied. The conservation relation (24) can be shown, adding the same assumptions as in the Lax Wendrof theorem, that the limit solution, if it exist, is a weak solution of (23), see [? ].…”

“…It is classically formulated for the simplified prototype (39). Here we will say that a scheme is positive, if in (39) we have c K ij ≥ 0, ∀i, j ∈ K and for all element K. In this sense, positivity can be shown to be equivalent to the so-called Local Extremum Diminishing property [22,23,24], and, provided that (39) is integrated with a SSP time marching scheme it leads to a discrete maximum principle, under a time step restriction. For example, when the explicit Euler scheme is used, one readily shows that…”

“…Detailed analysis of the accuracy of RD schemes, and the related constraints on the discretization can be found in [19,25] for the steady case, and in [26,24,10] for the time dependent case. In the P 1 case, the main result is that schemes admitting a set of uniformly bounded distribution coefficients are second order accurate.…”

Unsteady residual distribution schemes for transition prediction

“…This algorithm has a compact stencil (cell-based computations) and compute the residual over the cell using a finite element method. RDS were initially developed for the solution of scalar advection equations and subsequently extended to systems of equations (Deconinck and Ricchiuto [6]). Regardless of the temporal scheme used, the original RDS formulation cannot be more than first order accurate in timedependent computations due to the inconsistency of the spatial discretization (Ferrante and Deconinck [8]).…”