Fluctuation Splitting 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.

Fluctuation splitting Riemann solver for a non-conservative modeling of shear shallow water flow

In this paper we propose a fluctuation splitting finite volume scheme for a non-conservative modeling of shear shallow water flow (SSWF). This model was originally proposed by Teshukov (2007) in [14] and was extended to include modeling of friction by Gavrilyuk et al. (2018) in [7]. The directional splitting scheme proposed by Gavrilyuk et al. (2018) in [7] is tricky to apply on unstructured grids. Our scheme is based on the physical splitting in which we separate the characteristic waves of the model to form two different hyperbolic sub-systems. The fluctuations associated with each sub-systems are computed by developing Riemann solvers for these sub-systems in a local coordinate system. These fluctuations enable us to develop a Godunov-type scheme that can be easily applied on mixed/unstructured grids. While the equation of energy conservation is solved along with the SSWF model in [7], in this paper we solve only SSWF model equations.We develop a cell-centered finite volume code to validate the proposed scheme with the help of some numerical tests. As expected, the scheme shows first order convergence. The numerical simulation of 1D roll waves shows a good agreement with the experimental results. The numerical simulations of 2D roll waves show similar transverse wave structures as observed in [7].

Residual distribution schemes for Maxwell’s equations

This paper aims to deliver another approach of solving hyperbolic system of equation into the field of computational electromagnetics, known as the residual distribution (RD) or fluctuation splitting method. The RD scheme fills the interstice between finite-element method (FEM) and finite-volume method (FVM), where its idea of calculating flux residual imitates the FVM, but the numerical solutions within a discretized triangular mesh is interpolated with similitudes to the FEM. It was originally designed as a remedy to capture shock problem in Euler system under compact stencil, but its extension to linear-preserving scheme which is the main cause of this work, capable of giving second-order-accurate results and is congenial to the FEM framework. The RD scheme is always vertex-centered therefore has all the conserved variables E and H located at the same nodal point. This topology evades having both vertex-centered and cell-centered coordination in one single mesh, and permits time-discretization other than the staggered time marching scheme, such as the common backward-time discretization which is unconditionally stable. Another contribution of this work is to suggest that row-mass-lumping of the consistent mass-matrix would not mar too much the second-order-accuracy of LDA-RD scheme, which is an upwind scheme where the mass-matrix an impediment for time-updating during the last decade. A prior reconstruction of the upwind mass-matrix is done before lumping up the mass-matrix so that the time-marching nodal update is consistent with the distribution of mass-matrix. The current work covers 3 basic exercises of electromagnetics, they are 2D radiation problem, 2D scattering problem and 3D waveguide propagation. For the 2D scattering and 3D waveguide problems, the perfect electric conductor (PEC) boundary condition is successfully applied using RD construction.

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.

A Posteriori Analysis of a Positive Streamwise Invariant Discretization of a Convection-Diffusion Equation

We consider the finite element discretization of a convection-diffusion equation, where the convection term is handled via a fluctuation splitting algorithm. We prove a posteriori error estimates which allow us to perform mesh adaptivity in order to optimize the discretization of these equations. Numerical results confirm the interest of such an approach.

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.

Parallelizaton of visual magneto-hydrodynamics code based on fluctuation distribution scheme on triangular grids

A two-dimensional (2D) magneto-hydrodynamics (MHD) code which has visualization and parallel processing capability is presented in this paper. The code utilizes a fluctuation splitting (FS) scheme that runs on structured or unstructured triangular meshes. First FS scheme which included the wave model: Model-A had been developed by Roe [Roe PL. Discrete models for the numerical analysis of time-dependent multi-dimensional gas dynamics. J Comp Phys 1986;63:458–76.] for the solutions of Euler’s equations. The first 2D-MHD wave model: MHD-A, was then developed by Balci and Aslan [Balci Ş. The numerical solutions of two dimensional MHD equations by fluctuation splitting scheme on triangular meshes, Ph.D. Thesis, University of Marmara, Science-Art Faculty, Physics Dept Istanbul, Turkey; 2000; Aslan N. MHD-A: A fluctuation splitting wave model for planar magnetohydrodynamics. J Comp Phys 1999;153:437–66.] to solve MHD problems including shocks and discontinuities. It was then shown in [Balci S, Aslan N. Two dimensional MHD solver by fluctuation splitting and dual time stepping. Int J Numer Meth Fluids, in press.] that this code was capable of producing reliable results in compressible and nearly incompressible limits and under the effect of gravitational fields and that it was able to identically reduce to model-A of Roe in Euler limit with no sonic problems at rarefaction fans (Balci and Aslan, in press). An important feature of this code is its ability to run time dependent or steady problems on structured or unstructured triangular meshes that can be generated automatically by the code for specified domains. In order to use the parallel processing capability of the code, the triangular meshes are decomposed into different blocks in order to share the workload among a number of processors (here personal computers) which are connected by Ethernet. Due to the compact nature of the FS scheme, only one set of data transfer is required between neighbor processors. As it will be shown, this phenomenon results in minimum amount of communication loss and makes the scheme rather robust for parallel processing. The other important feature of the new code is its visual capability. As the code is running, colorful images of scalar quantities (density, pressure, Mach number, etc.) or vector graphics of vectoral quantities (velocity, magnetic field, etc.) can be followed on the screen. The extended code, called PV-MHDA, also allows following the trajectories of the particles in time by means of a recently included particle in cell (PIC) algorithm. Because the numerical dissipation embedded in its wave model reflects real physical viscosity and resistivity, it is able to run accurately for compressible flows (including shocks) as well as nearly incompressible flows (e.g., Kelvin–Helmholtz instability). The user-friendly visual and large-scale computation capability of the code allow the user more thorough analysis of MHD problems in two-dimensional complex domains.

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.

Two‐dimensional MHD solver by fluctuation splitting and dual time stepping

AbstractNumerical solutions of 2D magneto‐hydrodynamic (MHD) equations by means of a fluctuation splitting (FS) scheme (with a new wave model and dual time stepping technique) is presented. The FS scheme, essentially based on the model explained in Proceedings of the Tenth International Conference, vol. 10, Swansea, 21–25 July 1997; Godunov Symposium, University of Michigan, Ann Arbor, 1–2 May 1997; Physics Symposium, Alanya, Turkey, 27–31 October 1998; J. Comput. Phys. 1999; 153:437–466; Ph.D. Thesis, University of Marmara, Istanbul, Turkey, 2000), was extended to include gravitational source effects, limiters to limit oscillations, high order time accuracy through multistage Runge–Kutta steps, and a dual time stepping scheme to drive magnetic field divergence to zero during iterations. The numerical results show that with the new wave model called MHD‐B along with its embedded numerical dissipation, correct limiting viscosity solution has been recovered and that it can safely be used in order to investigate steady or time dependent magnetized or neutral compressible flows in two dimensions. Copyright © 2006 John Wiley & Sons, Ltd.

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.

A numerical scheme for ionizing shock waves

A two-dimensional (2D) visual computer code to solve the steady state (SS) or transient shock problems including partially ionizing plasma is presented. Since the flows considered are hypersonic and the resulting temperatures are high, the plasma is partially ionized. Hence the plasma constituents are electrons, ions and neutral atoms. It is assumed that all the above species are in thermal equilibrium, namely, that they all have the same temperature. The ionization degree is calculated from Saha equation as a function of electron density and pressure by means of a nonlinear Newton type root finding algorithms. The code utilizes a wave model and numerical fluctuation distribution (FD) scheme that runs on structured or unstructured triangular meshes. This scheme is based on evaluating the mesh averaged fluctuations arising from a number of waves and distributing them to the nodes of these meshes in an upwind manner. The physical properties (directions, strengths, etc.) of these wave patterns are obtained by a new wave model: ION-A developed from the eigen-system of the flux Jacobian matrices. Since the equation of state (EOS) which is used to close up the conservation laws includes electronic effects, it is a nonlinear function and it must be inverted by iterations to determine the ionization degree as a function of density and temperature. For the time advancement, the scheme utilizes a multi-stage Runge–Kutta (RK) algorithm with time steps carefully evaluated from the maximum possible propagation speed in the solution domain. The code runs interactively with the user and allows to create different meshes to use different initial and boundary conditions and to see changes of desired physical quantities in the form of color and vector graphics. The details of the visual properties of the code has been published before (see [N. Aslan, A visual fluctuation splitting scheme for magneto-hydrodynamics with a new sonic fix and Euler limit, J. Comput. Phys. 197 (2004) 1–27]). The two-dimensional nature of ION-A was presented by a planar shock wave propagating over a circular obstacle. It was demonstrated that including the effects of ionization in calculating complex flows is important, even when they appear initially negligible. This code can be used to accurately simulate the nonlinear time dependent evolution of neutral or ionized plasma flows from supersonic to hypersonic regimes.

An Implicit Fluctuation Splitting Scheme for Turbomachinery Flows

Abstract This paper describes an accurate, robust and efficient methodology for solving two-dimensional steady transonic turbomachinery flows. The Euler fluxes are discretized in space using a hybrid multidimensional upwind method, which, according to the local flow conditions, uses the most suitable fluctuation splitting (FS) scheme at each cell of the computational domain. The viscous terms are discretized using a standard Galerkin finite element scheme. The eddy viscosity is evaluated by means of the Spalart-Allmaras turbulence transport equation, which is discretized in space by means of a mixed FS-Galerkin approach. The equations are discretized in time using an implicit Euler scheme, the Jacobian being evaluated by two-point backward differences. The resulting large sparse linear systems are solved sequentially using a preconditioned GMRES strategy. The proposed methodology is employed to compute subsonic and transonic turbulent flows inside a high-turning turbine-rotor cascade.

A second-order-accurate monotone implicit fluctuation splitting scheme for unsteady problems

This paper provides an implicit fluctuation splitting scheme which achieves second-order accuracy in both space and time, while guaranteeing monotone solutions. The method is based on a dual-time-stepping approach which allows to separate the spatial and temporal discretizations. A novel distribution scheme of the unsteady-fluctuation term, coupled with a new limiting procedure for hampering spurious extrema, are the key-ingredients of the remarkable accuracy of the new method, which is applied with success to both scalar advection problems and unsteady inviscid flows in two space dimensions.

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.

A visual fluctuation splitting scheme for magnetohydrodynamics with a new sonic fix and Euler limit

This paper presents a two dimensional visual computer code developed to solve magnetohydrodynamic (MHD) equations. This code runs on structured and unstructured triangles and operates by a fluctuation splitting (FS) scheme. The FS scheme originally introduced by Roe [in: K.W. Morton, M.J. Baines (Eds.), Numerical Methods for Fluid Dynamics II, Academic Press, New York, 1982] to solve Euler equations was extended by Aslan [J. Comput. Phys. 153 (1999) 437] for solving ideal MHD equations. Aslan's method included a wave model, called MHD-A, consisting of slow and fast magneto-acoustic waves as well as an entropy and artificial magnetic monopole wave. In this work, Aslan's method was extended to include external sources, a new sonic fix, and a careful normalization in the Euler limit. It is shown by numerical experiments that VIS-MHD-A is able to work accurately for a wide range of problems including discontinuities, shock structures, and problems including smooth solutions (e.g., Rayleigh–Taylor and Kelvin–Helmholtz instability).

Adaptive grid generation by minimizing residuals

AbstractWe describe recent developments in the method of fluctuation splitting, in particular the minimization of residuals with respect to both the nodal values of the solution and also the nodal co‐ordinates. For elliptic problems, including external (lifting) flows, obtaining good results requires that the residuals be evaluated to third order. Solutions to typical airfoil problems can then be obtained with very economical grids. For hyperbolic problems, only a second‐order evaluation of the residual is needed. Shocks can be incorporated with arbitrarily high resolution by allowing elements to shrink to zero area, but to exclude the possibility of rarefaction shocks an adaptive quadrature must be employed. Copyright © 2002 John Wiley & Sons, Ltd.

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.

MHD-A: A Fluctuation Splitting Wave Model for Planar Magnetohydrodynamics

This paper describes a two-dimensional (2D) upwind residual distribution or fluctuation splitting (FS) scheme (MHD-A) for the numerical solutions of planar magnetohydrodynamics (MHD) equations on structured or unstructured triangular meshes. The scheme is second order in space and time, and utilizes a consistent 2D wave model originating from the eigensystem of a 2D jacobian matrix of the MHD flux vector. The possible waves existing in this wave model are entropy, magnetoacoustic, and (numerical) magnetic monopole waves; however, Alfven waves do not exist since the problem is planar.One of the important features of the method is that the mesh structure has no influence on propagation directions of the waves. These directions are dependent only on flow properties and field gradients (for example, it is shown that the magnetoacoustic waves propagate in the directions of maximum and minimum magnetic strain rates). The other feature is that no flux evaluations and no information from the neighboring cells are needed to obtain a second order, positive, and linearity preserving scheme.A variety of numerical tests carried out by the model on structured and unstructured triangular meshes show that MHD-A produces rather encouraging numerical results even though it is the first FS wave model ever developed for multidimensional MHD.