Fluctuation Splitting Schemes Research Articles

The Notion of Conservation for Residual Distribution Schemes (or Fluctuation Splitting Schemes), with Some Applications

In this paper, we discuss the notion of discrete conservation for hyperbolic conservation laws. We introduce what we call fluctuation splitting schemes (or residual distribution, also RDS) and show through several examples how these schemes lead to new developments. In particular, we show that most, if not all, known schemes can be rephrased in flux form and also how to satisfy additional conservation laws. This review paper is built on Abgrall et al. (Computers and Fluids 169:10–22, 2018), Abgrall and Tokareva (SIAM SISC 39(5):A2345–A2364, 2017), Abgrall (J Sci Comput 73:461–494, 2017), Abgrall (Methods Appl Math 18(3):327–351, 2018a) and Abgrall (J Comput Phys 372, 640–666, 2018b). This paper is also a direct consequence of the work of Roe, in particular Deconinck et al. (Comput Fluids 22(2/3):215–222, 1993) and Roe (J Comput Phys 43:357–372, 1981) where the notion of conservation was first introduced. In [26], Roe mentioned the Hermes project and the role of Dassault Aviation. Bruno Stoufflet, Vice President R&D and advanced business of this company, proposed me to have a detailed look at Deconinck et al. (Comput Fluids 22(2/3):215–222, 1993). To be honest, at the time, I did not understand anything, and this was the case for several years. I was lucky to work with Katherine Mer, who at the time was a postdoc, and is now research engineer at CEA. She helped me a lot in understanding the notion of conservation. The present contribution can be seen as the result of my understanding after many years of playing around with the notion of residual distribution schemes (or fluctuation-splitting schemes) introduced by Roe.

A mass-matrix formulation of unsteady fluctuation splitting schemes consistent with Roe’s parameter vector

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.

Residual distribution finite element method for convection-dominated problems

In this paper, we present a new stabilized finite element method for the simulation of nonlinear convection-dominated systems of PDEs. We seek to combine the robustness of upwind finite-volume methods for the discretization of strongly shocked flows with the ease and accuracy of C0 finite element reconstruction. This is achieved by exploiting the similarities between traditional SUPG method and finite volume residual distribution schemes. By constructing the finite element analog of finite-volume fluctuation-splitting schemes, we combine the ability to simulate PDEs with strongly discontinuous solutions of upwind finite-volume methods with the reliable accuracy of finite element reconstruction, which traditional finite-volume methods often lack on unstructured grids. Finally, we test the proposed algorithm with challenging problems drawn from hypersonic compressible flow and semiconductor device simulation.

Analysis of high-order-accurate fluctuation splitting schemes for steady advection

This article provides an analysis of high-order-accurate two-dimensional fluctuation splitting schemes for steady advection. Using Lagrangian elements, a residual distribution scheme is formulated and its properties are assessed. Distributing the residuals over the sub-triangles of each element allows one to obtain a well-posed scheme. It is also shown that the standard elemental approach is ill-posed, insofar as it produces an undetermined linear system. A steady scalar-advection problem is used to verify that the numerical schemes do obey the analysis.

Second-order-accurate explicit fluctuation splitting schemes for unsteady problems

This paper is concerned with the issue of obtaining explicit fluctuation splitting schemes which achieve second-order accuracy in both space and time on an arbitrary unstructured triangular mesh. A theoretical analysis demonstrates that, for a linear reconstruction of the solution, mass lumping does not diminish the accuracy of the scheme provided that a Galerkin space discretization is employed. Thus, two explicit fluctuation splitting schemes are devised which are second-order accurate in both space and time, namely, the well known Lax–Wendroff scheme and a Lax–Wendroff-type scheme using a three-point-backward discretization of the time derivative. A thorough mesh-refinement study verifies the theoretical order of accuracy of the two schemes on meshes with increasing levels of nonuniformity.

Non-oscillatory third order fluctuation splitting schemes for steady scalar conservation laws

This paper addresses the issue of constructing non-oscillatory, higher than second order, fluctuation splitting methods on unstructured triangular meshes. It highlights the reasons why existing approaches fail and proposes a procedure which can be applied to any high order fluctuation splitting scheme to impose positivity on it. Its success is demonstrated through application to a series of linear and nonlinear scalar problems, using a pseudo-time-stepping technique to reach steady state solutions on two-dimensional unstructured meshes.

Third-order-accurate fluctuation splitting schemes for unsteady hyperbolic problems

This paper provides a two-dimensional fluctuation splitting scheme for unsteady hyperbolic problems which achieves third-order accuracy in both space and time. For a scalar conservation law, the sufficient conditions for a stable fluctuation splitting scheme to achieve a prescribed order of accuracy in both space and time are derived. Then, using a quadratic space approximation of the solution over each triangular element, based on the reconstruction of the gradient at the three vertices, and a four-level backward discretization of the time derivative, an implicit third-order-accurate scheme is designed. Such a scheme is extended to the Euler system and is validated versus well-known scalar-advection problems and inviscid discontinuous flows.

Achieving high-order fluctuation splitting schemes by extending the stencil

An extension to the fluctuation splitting approach for approximating hyperbolic conservation laws is described, which achieves higher than second-order accuracy in both space and time by extending the range of the distribution of the fluctuations. Initial results are presented for a simple linear scheme which is third-order accurate in both space and time on uniform triangular grids. Numerically induced oscillations are suppressed by applying the flux-corrected transport algorithm. These schemes are evaluated in the context of existing fluctuation splitting approaches to modelling time-dependent flows and some suggestions for their future development are made.

Convergence of fluctuation-splitting schemes for two dimensional scalar conservation laws with a kinetic solver

The “fluctuation-splitting schemes” (FSS in short) have been introduced by Roe and Sildikover to solve advection equations on rectangular grids and then extended to triangular grids by Roe, Deconinck, Struij... For a two dimensional nonlinear scalar conservation law, we consider the case of a triangular grid and of a kinetic approach to reduce the discretization of the nonlinear equation to a linear equation and apply a particular FSS called N-scheme. We show that the resulting scheme converges strongly in \(L^2_{loc}\) in a finite volume sense.

Fluctuation Splitting Schemes for the Compressible and Incompressible Euler and Navier-Stokes Equations

Introduced in the late eighties by Roe, fluctuation splitting (or residual distribution) schemes have recently emerged as a viable alternative to Finite Volume and Finite Element methods for PDE based, fluid dynamics simulations using unstructured meshes. Their application to the numerical approximation of the compressible and incompressible Euler and Navier-Stokes equations is described, emphasizing low Mach number and incompressible applications. The advantages provided by time-preconditioning techniques are discussed and details of the implementation are given.

Diffusion Characteristics of Finite Volume and Fluctuation Splitting Schemes

The diffusive characteristics of two upwind schemes, multi-dimensional fluctuation splitting and locally one-dimensional finite volume, are compared for scalar advection–diffusion problems. Algorithms for the two schemes are developed for node-based data representation on median-dual meshes associated with unstructured triangulations in two spatial dimensions. Four model equations are considered: linear advection, non-linear advection, diffusion, and advection–diffusion, with cases chosen to mimic features present in compressible gas dynamics. Modular coding is employed to isolate the effects of the two approaches for upwind flux evaluation, allowing for head-to-head accuracy and efficiency comparisons. Both the stability of compressive limiters and the amount of artificial diffusion generated by the schemes are found to be grid-orientation dependent, with the fluctuation splitting scheme producing less artificial diffusion than the finite volume scheme. Convergence rates are compared for an advection–diffusion problem, with a speedup of 2.5 seen for fluctuation splitting versus finite volume when solved on the same mesh. However, accurate solutions to problems with small diffusion coefficients can be achieved on coarser meshes using fluctuation splitting rather than finite volume, so that when comparing convergence rates to reach a given accuracy, fluctuation splitting shows a speedup of 29 over finite volume for the test problem.

A critical analysis of multi-dimensional upwinding for the Euler equations

This paper provides a critical analysis and comparison of fluctuation splitting schemes applied to a linear scalar advection equation as well as to various simple wave decomposition models of the Euler equations. A theoretical analysis for the linear case and numerical experiments are presented. It is shown that compact-stencil fluctuation splitting schemes cannot provide second order accuracy either for nonhomogeneous scalar advection problems, or for general flows when combined with simple wave decompositions of the Euler equations.