A third-order semi-discrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems

Summary. We construct a new third-order semi-discrete genuinely multidimensional central scheme for systems of conservation laws and related convection-diffusion equations. This construction is based on a multidimensional extension of the idea, introduced in [17] – the use of more precise information about the local speeds of propagation, and integration over nonuniform control volumes, which contain Riemann fans. As in the one-dimensional case, the small numerical dissipation, which is independent of , allows us to pass to a limit as . This results in a particularly simple genuinely multidimensional semi-discrete scheme. The high resolution of the proposed scheme is ensured by the new two-dimensional piecewise quadratic non-oscillatory reconstruction. First, we introduce a less dissipative modification of the reconstruction, proposed in [29]. Then, we generalize it for the computation of the two-dimensional numerical fluxes. Our scheme enjoys the main advantage of the Godunov-type central schemes –simplicity, namely it does not employ Riemann solvers and characteristic decomposition. This makes it a universal method, which can be easily implemented to a wide variety of problems. In this paper, the developed scheme is applied to the Euler equations of gas dynamics, a convection-diffusion equation with strongly degenerate diffusion, the incompressible Euler and Navier-Stokes equations. These numerical experiments demonstrate the desired accuracy and high resolution of our scheme. Received February 7, 2000 / Published online December 19, 2000     

Asymptotic Expansions and Numerical Methods for Compressible and Low Mach Number Fluid Flow

The results of a formal asymptotic low Mach number analysis [5, 6] of the Euler equations of gas dynamics are used to extend the validity of a numerical method from the simulation of compressible inviscid flow fields to the low Mach number regime. Although, different strategies are applicable [7, 8, 5, 9] in this context we focus our view to a preconditioning technique recently proposed by Guillard and Viozat [16]. We present a finite volume approximation of the governing equations using a Lax‐Friedrichs scheme whereby a preconditioning of the incorporated numerical dissipation is employed. A discrete asymptotic analysis proves the validity of the scheme in the low Mach number regime.     

Une méthode numérique décentrée pour la résolution d'écoulements diphasiques

This Note focuses on the numerical approximation of two-fluid flow models described by six balance equations. We introduce an original splitting technique especially derived to use the approximate Riemann solvers of the usual gas dynamics and to allow for a straightforward extension to various and detailed exchange source terms. When based on suitable kinetic upwind schemes, the whole scheme preserves the positivity of all the thermodynamic variables under a fairly unrestrictive “CFL like” condition. Several stiff numerical teats, are presented including phase separation, in order to highlight the efficiency of the proposed method.     

High order upwind scheme for modelling turbulent shallow water flow in hydraulic structures

An unstructured finite volume model for quasi-2D free surface flow with wet-dry fronts and turbulence modelling is presented. The convective flux is discretised with either a an hybrid second-order/first-order scheme, or a fully second order scheme, both of them upwind Godunov's schemes based on Roe's average. The hybrid scheme uses a second order discretisation for the two unit discharge components, whilst keeping a first order discretisation for the water depth [2]. In such a way the numerical diffusion is much reduced, without a significant reduction on the numerical stability of the scheme, obtaining in such a way accurate and stable results. It is important to keep the numerical diffusion to a minimum level without loss of numerical stability, specially when modelling turbulent flows, because the numerical diffusion may interfere with the real turbulent diffusion. In order to avoid spurious oscillations of the free surface when the bathymetry is irregular, an upwind discretisation of the bed slope source term [4] with second order corrections is used [2]. In this way a fully second order scheme which gives an exact balance between convective flux and bed slope in the hydrostatic case is obtained. The k – ε equations are solved with either an hybrid or a second order scheme. In all the numerical simulations the importance of using a second order upwind spatial discretisation has been checked [1]. A first order scheme may give rather good predictions for the water depth, but it introduces too much numerical diffusion and therefore, it excessively smooths the velocity profiles. This is specially important when comparing different turbulence models, since the numerical diffusion introduced by a first order upwind scheme may be of the same order of magnitude as the turbulent diffusion. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Box-Line Relaxation Schemes for Solving the Steady Incompressible Navier-Stokes Equations Using Second-Order Upwind Differencing

1.Introducti0nThec0nvection-diffusionbehaviour0fthevisc0usincompressibleNavier-St0kesequati0nsisamains0urce0fdifficultiesinthenumericalsoluti0n.Whendiscretiz-ingtheequati0nsusingfinitedifferenceschemes,upwind0rhybridschemesareusuallyused0nthec0nvectiontermsforensuringthestability0fthediscretesystem[1].Thefirst-orderuPwinddifferencinghasprovedt0beinadequatef0rtheincompressibleNavier-St0kesequati0nswithlargeReynoldsnumbers,alth0ughtheresultingdiscretesystemsareverystableandeasilys0lved.In[5],…     

A difference scheme with anti-diffusion terms to improve the computation of contact discontinuities and the study of weak discontinuities in compressible flow

The investigation of Mach reflection formed after the impingement of a weak plane shock wave on a wedge with shock Mach number M_s near 1, is still an open problem[12]. It's difficult for shock tube experiments with interferometer to detect contact discontinuities if it is too weak; also difficult to catch with due accuracy the transition condition between Mach reflection and regular reflection. The interest to this phenomenon is continuing, especially for weak shocks, because there was systematic discrepancy between simplified three shock theory of von Neumann [8] and shock tube results [15] which was named by G. Birkhoff as “von Neumann Paradox on three shock theory” [18].In 1972, K.O.Friedrichs called for more computational efforts on this problem. Recently it is known that for weak impinging shocks it's still difficult to get contact discontinuities and curved Mach stem with satisfactory accuracy. Recent numerical computation sometimes even fails to show reflected shock wave[6]. These explain why von Neumann paradox of the three shock theory in case of weak discontinuities is still a problem of interesting [9,12,14]. In this paper, on one hand, we investigate the numerical methods for Euler's equation for compressible inviscid flow, aiming at improving the computation of contact discontinuities, on the other hand, a methodology is suggested to correctly plot flow data from the massive information in storage. On this basis, all the reflected shock wave , contact discontinuities and the curved Mach stem are determined. We get Mach reflection under the condition when over-simplified shock theory predicts no such configuration[5].     

An Upwind Difference Scheme and Its Corresponding Boundary Scheme

An upwind difference scheme was given by the author in [5] for the numerical solution of steady-state problems. The present work studies this upwind scheme and its corresponding boundary scheme for the numerical solution of unsteady problems. For interior points the difference equations are approximations of the characteristic relations; for boundary points difference equatons are approximations of the characteristicrelations corresponding to the outgoing characteristics and the "non-reflecting" boundary conditions. Calculation of a Riemann problem in a finite computational region yields promising numerical results.     

The relaxation schemes for systems of conservation laws in arbitrary space dimensions

We present a class of numerical schemes (called the relaxation schemes) for systems of conservation laws in several space dimensions. The idea is to use a local relaxation approximation. We construct a linear hyperbolic system with a stiff lower order term that approximates the original system with a small dissipative correction. The new system can be solved by underresolved stable numerical discretizations without using either Riemann solvers spatially or a nonlinear system of algebraic equations solvers temporally. Numerical results for 1-D and 2-D problems are presented. The second-order schemes are shown to be total variation diminishing (TVD) in the zero relaxation limit for scalar equations. ©1995 John Wiley & Sons, Inc.     

Comparison of the Solution of the Riemann Problem for Inviscid/Viscous Flows

The aim of this paper is to present the influence of the method used to solve the convective fluxes on the transonic internal flow field. The governing equations are discretized using an upwind method based on different solutions of the Riemann problem, flux vector splitting (FVS) or flux difference splitting schemes (FDS). Turbulence effects are simulated by means of the low‐Reynolds‐number k – ϵ and the SST (Shear‐Stress‐Transport) turbulence models.     

Kinetic approximation of a boundary value problem for conservation laws

Summary. We design numerical schemes for systems of conservation laws with boundary conditions. These schemes are based on relaxation approximations taking the form of discrete BGK models with kinetic boundary conditions. The resulting schemes are Riemann solver free and easily extendable to higher order in time or in space. For scalar equations convergence is proved. We show numerical examples, including solutions of Euler equations.Mathematics Subject Classification (2000): 65M06, 65M12, 76M20Correspondence to: D. Aregba-Driollet     

MUSTA: A multi-stage numerical flux

Numerical methods for conservation laws constructed in the framework of finite volume and discontinuous Galerkin finite elements require, as the building block, a monotone numerical flux. In this paper we present some preliminary results on the MUSTA approach [E.F. Toro, Multi-stage predictor–corrector fluxes for hyperbolic equations, Technical Report NI03037-NPA, Isaac Newton Institute for Mathematical Sciences, University of Cambridge, UK, 17th June, 2003] for constructing upwind numerical fluxes. The scheme may be interpreted as an un-conventional approximate Riemann solver that has simplicity and generality as its main features. When used in its first-order mode we observe that the scheme achieves the accuracy of the Godunov method used in conjunction with the exact Riemann solver, which is the reference first-order method for hyperbolic systems. At least for the scalar model hyperbolic equation, the Godunov scheme is the best of all first-order monote schemes, it has the smallest truncation error. Extensions of the scheme of this paper are realized in the framework of existing approaches. Here we present a second-order TVD (TVD for the scalar case) extension and show numerical results for the two-dimensional Euler equations on non-Cartesian geometries. The schemes find their best justification when solving very complex systems for which the solution of the Riemann problem, in the classical sense, is too complex, too costly or is simply unavailable.     

Central ADER schemes for hyperbolic conservation laws

In this paper we first briefly review the very high order ADER methods for solving hyperbolic conservation laws. ADER methods use high order polynomial reconstruction of the solution and upwind fluxes as the building block. They use a first order upwind Godunov and the upwind second order weighted average (WAF) fluxes. As well known the upwind methods are more accurate than central schemes. However, the superior accuracy of the ADER upwind schemes comes at a cost, one must solve exactly or approximately the Riemann problems (RP). Conventional Riemann solvers are usually complex and are not available for many hyperbolic problems of practical interest. In this paper we propose to use two central fluxes, instead of upwind fluxes, as the building block in ADER scheme. These are the monotone first order Lax-Friedrich (LXF) and the third order TVD flux. The resulting schemes are called central ADER schemes. Accuracy of the new schemes is established. Numerical implementations of the new schemes are carried out on the scalar conservation laws with a linear flux, nonlinear convex flux and non-convex flux. The results demonstrate that the proposed scheme, with LXF flux, is comparable to those using first and second order upwind fluxes while the scheme, with third order TVD flux, is superior to those using upwind fluxes. When compared with the state of art ADER schemes, our central ADER schemes are faster, more accurate, Riemann solver free, very simple to implement and need less computer memory. A way to extend these schemes to general systems of nonlinear hyperbolic conservation laws in one and two dimensions is presented.     

The upwind finite difference fractional steps method for combinatorial system of dynamics of fluids in porous media and its application

For combinatorial system of multilayer dynamics of fluids in porous media, the second order and first order upwind finite difference fractional steps schemes applicable to parallel arithmetic are put forward and two-dimensional and three-dimensional schemes are used to form a complete set. Some techniques, such as implicit-explicit difference scheme, calculus of variations, multiplicative commutation rule of difference operators, decomposition of high order difference operators and prior estimates, are adopted. Optimal order estimates in L ² norm are derived to determine the error in the second order approximate solution. This method has already been applied to the numerical simulation of migration-accumulation of oil resources. Keywords: combinatorial system, multilayer dynamics of fluids in porous media, two-class upwind finite difference fractional steps method, convergence, numerical simulation of energy sources.     

A Roe-type scheme with low Mach number preconditioning for a two-phase compressible flow model with pressure relaxation

We describe two-phase compressible flows by a hyperbolic six-equation single-velocity two-phase flow model with stiff mechanical relaxation. In particular, we are interested in the simulation of liquid-gas mixtures such as cavitating flows. The model equations are numerically approximated via a fractional step algorithm, which alternates between the solution of the homogeneous hyperbolic portion of the system through Godunov-type finite volume schemes, and the solution of a system of ordinary differential equations that takes into account the pressure relaxation terms. When used in this algorithm, classical schemes such as Roe’s or HLLC prove to be very efficient to simulate the dynamics of transonic and supersonic flows. Unfortunately, these methods suffer from the well known difficulties of loss of accuracy and efficiency for low Mach number regimes encountered by upwind finite volume discretizations. This issue is particularly critical for liquid-gasmixtures due to the large and rapid variation in the flow of the acoustic impedance. To cure the problem of loss of accuracy at low Mach number, in this work we apply to our original Roe-type scheme for the two-phase flow model the Turkel’s preconditioning technique studied by Guillard–Viozat [Computers & Fluids, 28, 1999] for the Roe’s scheme for the classical Euler equations.We present numerical results for a two-dimensional liquid-gas channel flow test that show the effectiveness of the resulting Roe-Turkel method for the two-phase system.     

Fine structures for the solutions of the two-dimensional Riemann problems by high-order WENO schemes

The two-dimensional Riemann problem with polytropic gas is considered. By a restriction on the constant states of each quadrant of the computational domain such that there is only one planar centered wave connecting two adjacent quadrants, there are nineteen genuinely different initial configurations of the problem. The configurations are numerically simulated on a fine grid and compared by the 5th-order WENO-Z5, 6th-order WENO-??6, and 7th-order WENO-Z7 schemes. The solutions are very well approximated with high resolution of waves interactions phenomena and different types of Mach shock reflections. Kelvin-Helmholtz instability-like secondary-scaled vortices along contact continuities are well resolved and visualized. Numerical solutions show that WENO-??6 outperforms the comparing WENO-Z5 and WENO-Z7 in terms of shock capturing and small-scaled vortices resolution. A catalog of the numerical solutions of all nineteen configurations obtained from the WENO-??6 scheme is listed. Thanks to their excellent resolution and sharp shock capturing, the numerical solutions presented in this work can be served as reference solutions for both future numerical and theoretical analyses of the 2D Riemann problem.     

Non-reflecting boundary conditions for the compressible Euler and Navier-Stokes equations in the finite volume method

The paper is concerned with the numerical implementation of the inlet and outlet boundary conditions in the finite volume method for the solution of the 3D Euler and Navier-Stokes equations. The explicit time marching procedure is described. The classical Riemann problem is modified for physically relevant boundary conditions with the aim to keep conservation laws. This technique was used in [2]. The initial condition in the Riemann problem is replaced by the suitable one-sided boundary condition. This results in the acceleration of the numerical method itself. On the inlet the pressure and the density and the angle of attack or velocity vector and the entropy are prescribed. On the outlet the pressure or normal component of the velocity or temperature or mass flow are investigated in such a way to obtain the unique solution of the modified Riemann problem. Various combinations of inlet and outlet boundary conditions are investigated. This results in the sufficiently precise approximation of real flow boundary conditions. Numerical examples illustrating the usefulness of the proposed approach for cascade flow are presented. Another numerical example is shown in [3]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The second‐order upwind finite difference fractional steps method for moving boundary value problem of oil‐water percolation

The research of the three‐dimensional (3D) compressible miscible (oil and water) displacement problem with moving boundary values is of great value to the history of oil‐gas transport and accumulation in basin evolution, as well as to the rational evaluation in prospecting and exploiting oil‐gas resources, and numerical simulation of seawater intrusion. The mathematical model can be described as a 3D‐coupled system of nonlinear partial differential equations with moving boundary values. For a generic case of 3D‐bounded region, a kind of second‐order upwind finite difference fractional steps schemes applicable to parallel arithmetic is put forward. Some techniques, such as the change of variables, calculus of variations, and the theory of a priori estimates, are adopted. Optimal order estimates in l² norm are derived for the errors in approximate solutions. The research is important both theoretically and practically for model analysis in the field, for model numerical method and for software development. Thus, the well‐known problem has been solved.Copyright © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1103–1129, 2014     

Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory

We consider asymptotics for orthogonal polynomials with respect to varying exponential weights w_n(x)dx = e^−nV(x) dx on the line as n → ∞. The potentials V are assumed to be real analytic, with sufficient growth at infinity. The principle results concern Plancherel‐Rotach‐type asymptotics for the orthogonal polynomials down to the axis. Using these asymptotics, we then prove universality for a variety of statistical quantities arising in the theory of random matrix models, some of which have been considered recently in [31] and also in [4]. An additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials (see also [4]). The orthogonal polynomial problem is formulated as a Riemann‐Hilbert problem following [19, 20]. The Riemann‐Hilbert problem is analyzed in turn using the steepest‐descent method introduced in [12] and further developed in [11, 13]. A critical role in our method is played by the equilibrium measure dμ_V for V as analyzed in [8]. © 1999 John Wiley & Sons, Inc.     

Application of MFCAV Riemann Solver to Maire's Cell-Centered Lagrangian Method

In this paper, we apply arbitrary Riemann solvers, which may not satisfy the Maire's requirement, to the Maire's node-based Lagrangian scheme developed in [P. H. Maire et al., SIAM J. Sci. Comput, 29 (2007), 1781-1824]. In particular, we apply the so-called Multi-Fluid Channel on Averaged Volume (MFCAV) Riemann solver and a Riemann solver that adaptively combines the MFCAV solver with other more dissipative Riemann solvers to the Maire's scheme. It is noted that neither of the two solvers satisfies the Maire's requirement. Numerical experiments are presented to demonstrate that the application of the two Riemann solvers is successful.