Roe Matrix Research Articles

Analysis of heterogeneous vehicular traffic: Using proportional densities

This paper presents a multi-class Aw–Rascle (AR) model with area occupancy expressed in terms of vehicle class proportions. Two vehicle classes that is; cars and motorcycles are considered based on an assumption that proportions of these form total traffic density. Qualitative properties of the proposed equilibrium velocity are established. Conditions under which the proposed model is stable are determined by linear stability analysis. To compute numerical flux, the model is decomposed by original Roe decomposition, where Roe matrix, averaged data variables and wave strengths are explicitly derived. The Roe matrix is shown to be hyperbolic, consistent and conservative. From the numerical results, the effect of proportional densities on the flow of vehicle classes is determined. Results obtained remain within limits therefore, the proposed model is realistic.

On the efficient implementation of PVM methods and simple Riemann solvers. Application to the Roe method for large hyperbolic systems

Polynomial Viscosity Matrix (PVM) methods can be considered as approximations of the Roe method in which the absolute value of the Roe matrix appearing in the numerical viscosity is replaced by the evaluation of the Roe matrix at a chosen polynomial that approximates the absolute value function. They are in principle cheaper than the Roe method since the computation and the inversion of the eigenvector matrix is not necessary. In this article, an efficient implementation of the PVM based on polynomials that interpolate the absolute value function at some points is presented. This implementation is based on the Newton form of the polynomials. Moreover, many numerical methods based on simple Riemann solvers (SRS) may be interpreted as PVM methods and thus this implementation can be also applied to them: the close relation between PVM methods and simple Riemann solvers is revisited here and new shorter proofs based on the classical interpolation theory are given. In particular, Roe method can be interpreted both as a SRS and as a PVM method so that the new implementation can be used. This implementation, that avoids the computation and the inversion of the eigenvector matrix, is called Newton Roe method. Newton Roe method yields the same numerical results of the standard Roe method, with less runtime for large PDE systems. Numerical results for two-layer Shallow Water Equations and Quadrature-Based Moment Equations show a significant speedup if the number of equations is large enough.

Polynomial viscosity methods for multispecies kinematic flow models

Multispecies kinematic flow models are defined by systems of strongly coupled, nonlinear first‐order conservation laws. They arise in various applications including sedimentation of polydisperse suspensions and multiclass vehicular traffic. Their numerical approximation is a challenge since the eigenvalues and eigenvectors of the corresponding flux Jacobian matrix have no closed algebraic form. It is demonstrated that a recently introduced class of fast first‐order finite volume solvers, called polynomial viscosity matrix (PVM) methods [M. J. Castro Díaz and E. Fernández‐Nieto, SIAM J Sci Comput 34 (2012), A2173–A2196], can be adapted to multispecies kinematic flows. PVM methods have the advantage that they only need some information about the eigenvalues of the flux Jacobian, and no spectral decomposition of a Roe matrix is needed. In fact, the so‐called interlacing property (of eigenvalues with known velocity functions), which holds for several important multispecies kinematic flow models, provides sufficient information for the implementation of PVM methods. Several variants of PVM methods (differing in polynomial degree and the underlying quadrature formula to approximate the Roe matrix) are compared by numerical experiments. It turns out that PVM methods are competitive in accuracy and efficiency with several existing methods, including the Harten, Lax, and van Leer method and a spectral weighted essentially non‐oscillatory scheme that is based on the same interlacing property. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1265–1288, 2016

A second order PVM flux limiter method. Application to magnetohydrodynamics and shallow stratified flows

In this work we propose a second order flux limiter finite volume method, named PVM-2U-FL, that only uses information of the two external waves of the hyperbolic system. This method could be seen as a natural extension of the well known WAF method introduced by E.F. Toro in [23]. We prove that independently of the number of unknowns of the 1D system, it recovers the second order accuracy at regular zones, while in presence of discontinuities, the scheme degenerates to PVM-2U method, which can be seen as an improvement of the HLL method (see [6,10]). Another interesting property of the method is that it does not need any spectral decomposition of the Jacobian or Roe matrix associated to the flux function. Therefore, it can be easily applied to systems with a large number of unknowns or in situations where no analytical expression of the eigenvalues or eigenvectors are known. In this work, we apply the proposed method to magnetohydrodynamics and to stratified multilayer flows. Comparison with the two-waves WAF and HLL-MUSCL methods are also presented. The numerical results show that PVM-2U-FL is the most efficient and accurate among them.

WELL-BALANCED ROE-TYPE NUMERICAL SCHEME FOR A MODEL OF TWO-PHASE COMPRESSIBLE FLOWS

We present a multi-stage Roe-type numerical scheme for a model of two-phase flows arisen from the modeling of deflagration-to-detonation transition in granular materials. The first stage in the construction of the scheme computes the volume fraction at every time step. The second stage deals with the nonconservative terms in the governing equations which produces states on both side of the contact wave at each node. In the third stage, a Roe matrix for the two-phase is used to apply on the states obtained from the second stage. This scheme is shown to capture stationary waves and preserves the positivity of the volume fractions. Finally, we present numerical tests which all indicate that the proposed scheme can give very good approximations to the exact solution.

A Class of Computationally Fast First Order Finite Volume Solvers: PVM Methods

In this work, we present a class of fast first order finite volume solvers, called PVM (polynomial viscosity matrix), for balance laws or, more generally, for nonconservative hyperbolic systems. They are defined in terms of viscosity matrices computed by a suitable polynomial evaluation of a Roe matrix. These methods have the advantage that they only need some information about the eigenvalues of the system to be defined, and no spectral decomposition of a Roe matrix is needed. As a consequence, they are faster than the Roe method. These methods can be seen as a generalization of the schemes introduced by Degond et al. in [C. R. Acad. Sci. Paris Sér. l Math., 328 (1999), pp. 479--483] for balance laws and nonconservative systems. The first order path conservative methods to be designed here are intended to be used as the basis for higher order methods for multidimensional problems. In this work, some well known solvers, such as Rusanov, Lax--Friedrichs, FORCE (see [E. F. Toro and S. J. Billett, IMA J. Numer. Anal., 20 (2000), pp. 47--79; M. J. Castro, A. Pardo, C. Parés, and E. F. Toro, Math. Comp., 79 (2010), pp. 1427--1472]), GFORCE (see [E. F. Toro and V. A. Titarev, J. Comput. Phys., 216 (2006), pp. 403--429]), and HLL (see [A. Harten, P. D. Lax, and B. van Leer, SIAM Rev., 25 (1983), pp. 35--61]), are redefined under this form, and then some new solvers are proposed. Finally, some numerical tests are presented, and the performances of the numerical schemes are compared with each others and with Roe scheme for the 1D and 2D two-fluid flow model of Pitman and Le.

Application of point implicit Runge–Kutta methods to inviscid and laminar flow problems using AUSM and AUSM+ upwinding

Explicit Runge–Kutta methods preconditioned by a pointwise matrix valued preconditioner can significantly improve the convergence rate to approximate steady state solutions of laminar flows. This has been shown for central discretisation schemes and Roe upwinding. Since the first-order approximation to the inviscid flux assuming constant weighting of the dissipative terms is given by the absolute value of the Roe matrix, the construction of the preconditioner is rather simple compared to other upwind techniques. However, in this article we show that similar improvements in the convergence rates can also be obtained for the AUSM+ scheme. Following the ideas for the central and Roe schemes, the preconditioner is obtained by a first-order approximation to the derivative of the convective flux. Viscous terms are included into the preconditioner considering a thin shear layer approximation. A complete derivation of the derivative terms is shown. In numerical examples, we demonstrate the improved convergence rates when compared with a standard explicit Runge–Kutta method accelerated with local time stepping.

On an Intermediate Field Capturing Riemann Solver Based on a Parabolic Viscosity Matrix for the Two-Layer Shallow Water System

The goal of this article is to design a new approximate Riemann solver for the two-layer shallow water system which is fast compared to Roe schemes and accurate compared to Lax-Friedrichs, FORCE, or GFORCE schemes (see Castro et al. in Math. Comput. 79:1427–1472, 2010). This Riemann solver is based on a suitable decomposition of a Roe matrix (see Toumi in J. Comput. Phys. 102(2):360–373, 1992) by means of a parabolic viscosity matrix (see Degond et al. in C. R. Acad. Sci. Paris 1 328:479–483, 1999) that captures some information concerning the intermediate characteristic fields. The corresponding first order numerical scheme, which is called IFCP (Intermediate Field Capturing Parabola) is linearly L ∞-stable, well-balanced, and it doesn’t require an entropy-fix technique. Some numerical experiments are presented to compare the behavior of this new scheme with Roe and GFORCE methods.

A Godunov-type Finite Volume Scheme for Meso- and Micro-scale Flows in Three Dimensions

This short note reports the extension of the f-waves approximate Riemann solver (Ahmad and Lindeman, 2007; LeVeque, 2002; Bale et al., 2002) for three-dimensional meso- and micro-scale atmospheric flows. The Riemann solver employs flux-based wave decomposition for the calculation of Godunov fluxes and does not require the explicit definition of the Roe matrix to enforce conservation. The other important feature of the Riemann solver is its ability to incorporate source term due to gravity without introducing discretization errors. The resulting finite volume scheme is second-order accurate in space and time. The finite-difference schemes currently used in atmospheric flow models are neither conservative nor able to resolve regions of sharp gradients. The finite volume scheme described in this paper is fully conservative and has the ability to resolve regions of sharp gradients without introducing spurious oscillations in the solution. The scheme shows promise in accurately resolving flows on the meso- and micro-scales and should be considered for implementation in the dynamical cores of next generation meso- and micro-scale atmospheric flow models.

Numerical solution of axisymmetric multi-species compressible gas flow: towards improved circuit breaker simulation

Nozzle wall ablation caused by high-temperature electrical arcs is studied in the context of high voltage SF6 circuit breakers. The gases generated by the ablation mix with insulating gas, thereby modifying the thermodynamics and chemistry of the problem. To simulate these phenomena, an axisymmetric Euler equations model for multi-species flow at local thermodynamic equilibrium has been developed. The governing equations are solved using a finite-volume method based on Roe's flux-splitting scheme and a new procedure is proposed to obtain the mean values used in Roe's matrix involving multi-species. The change in the gas composition due to dissociation, ionisation and recombinations are taken into account by a dynamic coupling to a thermodynamic database. The formulation is appropriate for general equations of state. The scheme has been verified by comparisons with the analytical solution of a classic shock-tube problem. An experimental validation is presented involving the simulation of shock interactions with helium cylinders and bubbles in air. The scheme is also compared to experimental ablation results, as well as a previous version of the code. The model is then applied to a well controlled experimental arrangement with an arc inside a small Teflon® tube. Finally, a simulation with a model circuit breaker completes the study.

The approximate Riemann solver of Roe applied to a drift-flux two-phase flow model

We construct a Roe-type numerical scheme for approximating the solutions of a drift- flux two-phase flow model. The model incorporates a set of highly complex closure laws, and the fluxes are generally not algebraic functions of the conserved variables. Hence, the classical approach of constructing a Roe solver by means of parameter vectors is unfeasible. Alternative approaches for analytically constructing the Roe solver are discussed, and a formulation of the Roe solver valid for general closure laws is derived. In particular, a fully analytical Roe matrix is obtained for the special case of the Zuber-Findlay law describing bubbly flows. First and second-order accurate versions of the scheme are demonstrated by numerical examples.

Numerical method of the Riemann problem for two-dimensional multi-fluid flows with general equation of state

Based on the classical Roe method, we develop an interface capture method according to the general equation of state and extend the single-fluid Roe method to the two-dimensional (2D) multi-fluid flows, as well as construct the continuous Roe matrix for the whole flow field. The interface capture equations and fluid dynamic conservative equations are coupled together and solved by using any high-resolution schemes that usually suit for the single-fluid flows. Some numerical examples are given to illustrate the solution of 1D and 2D multi-fluid Riemann problems.

High-resolution FEM–FCT schemes for multidimensional conservation laws

The flux-corrected-transport paradigm is generalized to implicit finite element schemes and nonlinear systems of hyperbolic conservation laws. In the scalar case, a nonoscillatory low-order method of upwind type is derived by elimination of negative off-diagonal entries of the discrete transport operator. The difference between the discretizations of high and low order is decomposed into a sum of skew-symmetric antidiffusive fluxes. An iterative flux limiter independent of the time step is proposed for implicit schemes. The nonlinear antidiffusion is incorporated into the solution in the framework of a defect correction scheme preconditioned by the monotone low-order operator. In the case of a hyperbolic system, the global Jacobian matrix is assembled edge-by-edge without resorting to numerical integration. Its low-order counterpart is constructed by rendering all off-diagonal blocks positive definite or adding scalar artificial diffusion proportional to the spectral radius of the Roe matrix. The coupled equations are solved in a segregated manner within an outer defect correction loop equipped with a block-diagonal preconditioner. After a suitable synchronization, the correction factors evaluated for an arbitrary set of indicator variables are applied to the antidiffusive fluxes which are inserted into the global defect vector. The performance of the new algorithm is illustrated by numerical examples for scalar transport problems and the compressible Euler equations.

Solution of the hyperbolic mild‐slope equation using the finite volume method

AbstractA finite volume solver for the 2D depth‐integrated harmonic hyperbolic formulation of the mild‐slope equation for wave propagation is presented and discussed. The solver is implemented on unstructured triangular meshes and the solution methodology is based upon a Godunov‐type second‐order finite volume scheme, whereby the numerical fluxes are computed using Roe's flux function. The eigensystem of the mild‐slope equations is derived and used for the construction of Roe's matrix. A formulation that updates the unknown variables in time implicitly is presented, which produces a more accurate and reliable scheme than hitherto available. Boundary conditions for different types of boundaries are also derived. The agreement of the computed results with analytical results for a range of wave propagation/transformation problems is very good, and the model is found to be virtually paraxiality‐free. Copyright © 2003 John Wiley & Sons, Ltd.

A Roe scheme for the Bi-temperature model of magnetohydrodynamics

In this paper, a Roe scheme for the bi-temperature magnetohydrodynamics (MHD) model is set up. A Roe matrix is obtained for the one-dimensional system in Eulerian coordinates. These results are extended to the two-dimensional case. One- and two-dimensional numerical examples are provided, showing Roe solver efficiency.

Roe Matrices for Ideal MHD and Systematic Construction of Roe Matrices for Systems of Conservation Laws

In this paper, the construction of a Roe's scheme for the conservative system of ideal magnetohydrodynamics (MHD) is presented. As this method relies on the computation of a Roe matrix, the problem is to find a matrixA(Ul,Ur) which satisfies the following properties. It is required to be consistent with the jacobian of the fluxF, to have real eigenvalues, a complete set of eigenvectors and to satisfy the relation: ΔF=A(Ul,Ur) ΔU, whereUlandUrare two admissible states and ΔUtheir difference. For the ideal MHD system, using eulerian coordinates, a Roe matrix is obtained without any hypothesis on the specific heat ratio. Especially, its construction relies on an original expression of the magnetic pressure jump. Moreover, a Roe matrix is computed for lagrangian ideal MHD, by extending the results of Munz who obtained such a matrix for the system of lagrangian gas dynamics. So this second matrix involves arithmetic averages unlike the eulerian one, which contains classical Roe averages like in eulerian gas dynamics. In this paper, a systematic construction of lagrangian Roe matrices in terms of eulerian Roe matrices for a general system of conservation laws is also presented. This result, applied to the above eulerian and lagrangian matrices for ideal MHD, gives two new matrices for this system. In the same way, by applying this construction to the gas dynamics equations new Roe matrices are also obtained. All these matrices allow the construction of Roe type schemes. Some numerical examples on the shock tube problem show the applicability of this method.

Assessment of preconditioning methods for multidimensional aerodynamics

We consider the steady state equations for a compressible fluid. For low speed flow the system is stiff since the ratio of the convective speed to the speed of sound is very small. To overcome this difficulty we alter the time dependency of the equations while retaining the same steady state operator. In order to achieve high numerical resolution we also alter the artificial dissipation (or Roe matrix) of the numerical scheme. The definition of preconditioners and artificial dissipation terms can be formulated conveniently by using other sets of dependent variables rather than the conservation variables. The effects of different preconditioners, artificial dissipation and grid density on accuracy and convergence to the steady state of the numerical solutions are presented in detail. The numerical results obtained for inviscid and viscous two- and three-dimensional flows over external aerodynamic bodies indicate that efficient multigrid computations of flows with very low Mach numbers are now possible.

An upwind differencing scheme for the equations of ideal magnetohydrodynamics

Recently, upwind differencing schemes have become very popular for solving hyperbolic partial differential equations, especially when discontinuities exist in the solutions. Among many upwind schemes successfully applied to the problems in gas dynamics, Roe's method stands out for its relative simplicity and clarity of the underlying physical model. In this paper, an upwind differencing scheme of Roe-type for the MHD equations is constructed. In each computational cell, the problem is first linearized around some averaged state which preserves the flux differences. Then the solution is advanced in time by computing the wave contributions to the flux at the cell interfaces. One crucial task of the linearization procedure is the construction of a Roe matrix. For the special case γ = 2, a Roe matrix in the form of a mean value Jacobian is found, and for the general case, a simple averaging procedure is introduced. All other necessary ingredients of the construction, which include eigenvalues, and a complete set of right eigenvectors of the Roe matrix and decomposition coefficients are presented. As a numerical example, we chose a coplanar MHD Riemann problem. The problem is solved by the newly constructed second-order upwind scheme as well as by the Lax-Friedrichs, the Lax-Wendroff, and the flux-corrected transport schemes. The results demonstrate several advantages of the upwind scheme. In this paper, we also show that the MHD equations are nonconvex. This is a contrast to the general belief that the fast and slow waves are like sound waves in the Euler equations. As a consequence, the wave structure becomes more complicated; for example, compound waves consisting of a shock and attached to it a rarefaction wave of the same family can exist in MHD.