High-order Finite Volume Research Articles

High-order well-balanced finite-volume schemes for barotropic flows: Development and numerical comparisons

In this paper, we compare a classical finite-difference and a high order finite-volume scheme for barotropic ocean flows. We compare the schemes with respect to their accuracy, stability, and study various outflow and inflow boundary conditions. We apply the schemes to the problem of eddy formation in shelf slope jets along the Ormen Lange section of the Norwegian shelf. Our results strongly confirm the development of mesoscale eddies caused by instability of the flows.

An implicit high order finite volume scheme for the solution of 3D Navier–Stokes equations with new discretization of diffusive terms

The computation of three-dimensional viscous flows on complex geometries requiring distorted meshes is of great interest. This paper presents a finite volume solver using a quadratic reconstruction of the unknowns for the advective fluxes computation, and a conservative and consistent discretization of the diffusive terms, based on an extended version of the Coirier's diamond path. A fully implicit time integration procedure is employed with a preconditioned matrix-free GMRES solver.

An implicit finite volume method for the solution of 3D low Mach number viscous flows using a local preconditioning technique

This paper presents a cell-centered high order finite volume scheme for the solution of the three-dimensional (3D) Navier–Stokes equations with low Mach number. The system of non-linear equations is solved by means of a fully implicit pseudo-transient scheme. Each pseudo-time step is solved by a Newton-GMRes procedure. A local preconditioning technique is used to scale the speed of sound and to improve the system condition number for low Mach number and low cell Reynolds number. This preconditioning is applied to the AUSM+up flux vector splitting function. The method is tested on 2D and 3D low Mach number laminar flows.

A spectral finite volume transport scheme on the cubed-sphere

Advective processes are of central importance in many applications and their treatment is crucial in the numerical modelling of the transport of trace constituents in atmospheric models. High-order numerical methods offer the promise of accurately capturing these advective processes in atmospheric flows and have been shown to efficiently scale to large numbers of processors. In this paper, a conservative transport scheme based on the nodal high-order spectral finite volume method is developed for the cubed-sphere. A third-order explicit strong stability preserving scheme is employed for the time integration. The reconstruction procedure which we developed avoids the (expensive) calculation of the inverse of the reconstruction matrix. Flux-corrected transport algorithm is implemented to enforce monotonicity in the two-dimensional transport scheme. Two standard advection tests, a solid-body rotation and a deformational flow, were performed to evaluate the spectral finite volume method optionally combined with a flux-corrected transport scheme. Spectral accuracy in space is demonstrated with a linear wave equation.

The numerical prediction of planar viscoelastic contraction flows using the pom–pom model and higher-order finite volume schemes

This study investigates the numerical solution of viscoelastic flows using two contrasting high-order finite volume schemes. We extend our earlier work for Poiseuille flow in a planar channel and the single equation form of the extended pom–pom (SXPP) model [M. Aboubacar, J.P. Aguayo, P.M. Phillips, T.N. Phillips, H.R. Tamaddon-Jahromi, B.A. Snigerev, M.F. Webster, Modelling pom–pom type models with high-order finite volume schemes, J. Non-Newtonian Fluid Mech. 126 (2005) 207–220], to determine steady-state solutions for planar 4:1 sharp contraction flows. The numerical techniques employed are time-stepping algorithms: one of hybrid finite element/volume type, the other of pure finite volume form. The pure finite volume scheme is a staggered-grid cell-centred scheme based on area-weighting and a semi-Lagrangian formulation. This may be implemented on structured or unstructured rectangular grids, utilising backtracking along the solution characteristics in time. For the hybrid scheme, we solve the momentum-continuity equations by a fractional-staged Taylor–Galerkin pressure-correction procedure and invoke a cell-vertex finite volume scheme for the constitutive law. A comparison of the two finite volume approaches is presented, concentrating upon the new features posed by the pom–pom class of models in this context of non-smooth flows. Here, the dominant feature of larger shear and extension in the entry zone influences both stress and stretch, so that larger stretch develops around the re-entrant corner zone as Weissenberg number increases, whilst correspondingly stress levels decline.

A weighted least square scheme for compressible flows

The article describes the development of a high order finite volume method for the solution of transonic flow problems. The method is based on a reconstruction procedure similar to the weighted essentially non-oscillatory scheme (WENO). The analysis of accuracy and stability of the method is carried out for the case of smooth data and for simple discontinuity. The computational results demonstrate the performance of the WLSQR method for the solution of several flow problems in 2D and 3D using both structured and unstructured meshes.

High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems

This paper is concerned with the development of high order methods for the numerical approximation of one-dimensional nonconservative hyperbolic systems. In particular, we are interested in high order extensions of the generalized Roe methods introduced by I. Toumi in 1992, based on WENO reconstruction of states. We also investigate the well-balanced properties of the resulting schemes. Finally, we will focus on applications to shallow-water systems.

Staggered Finite Difference Schemes for Conservation Laws

In this work, we introduce new finite-difference shock-capturing central schemes on staggered grids. Staggered schemes may have better resolution of the corresponding unstaggered schemes of the same order. They are based on high-order nonoscillatory reconstruction (ENO or WENO), and a suitable ODE solver for the computation of the integral of the flux. Although they suffer from a more severe stability restriction, they do not require a numerical flux function. A comparison of the new schemes with high-order finite volume (on staggered and unstaggered grids) and high order unstaggered finite difference methods is reported.

MUSTA fluxes for systems of conservation laws

This paper is about numerical fluxes for hyperbolic systems and we first present a numerical flux, called GFORCE, that is a weighted average of the Lax-Friedrichs and Lax-Wendroff fluxes. For the linear advection equation with constant coefficient, the new flux reduces identically to that of the Godunov first-order upwind method. Then we incorporate GFORCE in the framework of 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], resulting in a version that we call GMUSTA. For non-linear systems this gives results that are comparable to those of the Godunov method in conjunction with the exact Riemann solver or complete approximate Riemann solvers, noting however that in our approach, the solution of the Riemann problem in the conventional sense is avoided. Both the GFORCE and GMUSTA fluxes are extended to multi-dimensional non-linear systems in a straightforward unsplit manner, resulting in linearly stable schemes that have the same stability regions as the straightforward multi-dimensional extension of Godunov’s method. The methods are applicable to general meshes. The schemes of this paper share with the family of centred methods the common properties of being simple and applicable to a large class of hyperbolic systems, but the schemes of this paper are distinctly more accurate. Finally, we proceed to the practical implementation of our numerical fluxes in the framework of high-order finite volume WENO methods for multi-dimensional non-linear hyperbolic systems. Numerical results are presented for the Euler equations and for the equations of magnetohydrodynamics.

Arbitrary high order finite-volume methods for electromagnetic wave propagation

Problems in electromagnetic wave propagation often require high accuracy approximations with low resolution computational grids. For non-stationary problems such schemes should possess the same approximation order in space and time. In the present article we propose for electromagnetic applications an explicit class of robust finite-volume ( FV) schemes for the Maxwell equations. To achieve high accuracy we combine the FV method with the so-called ADER approach resulting in schemes which are arbitrary high order accurate in space and time. Numerical results and convergence investigations are shown for two and three-dimensional test cases on Cartesian grids, where the used FV-ADER schemes are up to 8th order accurate in both space and time.

A fourth-order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq-type equations. Part I: model development and analysis

A high-order finite volume scheme is developed to numerically integrate a fully nonlinear and weakly dispersive set of Boussinesq-type equations (the so-called Serre equations) (J. Fluid Mech. 1987; 176:117–134; Surveys Geophys. 2004; 25(3–4):315–337). The choice of this discretization strategy is motivated by the fact that this particular set of equations is recasted in a convenient quasi-conservative form. Cell face values are reconstructed using implicit compact schemes (J. Comput. Phys. 1999; 156:137–180; J. Comput. Phys. 2004; 198:535–566) and time integration is performed with the help of a four-stage Runge–Kutta method. Numerical properties of the proposed scheme are investigated both, analytically using linear spectral analysis, and numerically for highly nonlinear cases. The numerical analysis indicates that the newly developed scheme has wider stability regions and better spectral resolution than most of the previously published numerical methods used to handle equivalent set of equations. Moreover, it was also noticed that the use of mixed-order strategies to discretize convective and dispersive terms may result in an important overall reduction of the spectral resolution of the scheme. Additionally, there is some numerical evidence, which seems to indicate that the incorporation of a high-order dispersion correction term as given by Madsen et al. (Coastal Eng. 1991; 15:371–388) may introduce instability in the system. Copyright © 2006 John Wiley & Sons, Ltd.

High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms

Hyperbolic balance laws have steady state solutions in which the flux gradients are nonzero but are exactly balanced by the source term. In our earlier work [J. Comput. Phys. 208 (2005) 206–227; J. Sci. Comput., accepted], we designed a well-balanced finite difference weighted essentially non-oscillatory (WENO) scheme, which at the same time maintains genuine high order accuracy for general solutions, to a class of hyperbolic systems with separable source terms including the shallow water equations, the elastic wave equation, the hyperbolic model for a chemosensitive movement, the nozzle flow and a two phase flow model. In this paper, we generalize high order finite volume WENO schemes and Runge–Kutta discontinuous Galerkin (RKDG) finite element methods to the same class of hyperbolic systems to maintain a well-balanced property. Finite volume and discontinuous Galerkin finite element schemes are more flexible than finite difference schemes to treat complicated geometry and adaptivity. However, because of a different computational framework, the maintenance of the well-balanced property requires different technical approaches. After the description of our well-balanced high order finite volume WENO and RKDG schemes, we perform extensive one and two dimensional simulations to verify the properties of these schemes such as the exact preservation of the balance laws for certain steady state solutions, the non-oscillatory property for general solutions with discontinuities, and the genuine high order accuracy in smooth regions.

Modelling pom-pom type models with high-order finite volume schemes

We are concerned with the numerical solution of viscoelastic flows using two contrasting high-order finite volume schemes. We take our earlier work for transient start-up flow in a channel and extend this beyond Oldroyd-B modelling to consider a different fluid model of the pom-pom class. This includes Single Extended form of the pom-pom model (SXPP), comparing the results of two different finite volume schemes. The numerical techniques employed are time-stepping algorithms, one of hybrid finite element/volume type, the other of pure finite volume form. The pure finite volume scheme is a staggered-grid cell-centred scheme based on area-weighting and a semi-Lagrangian formulation. This may be implemented on structured or unstructured rectangular grids, utilising backtracking along the solution characteristics in time. For the hybrid scheme, we solve the momentum/continuity equations by a fractional-staged Taylor–Galerkin/pressure-correction procedure and invoke a cell-vertex finite volume scheme for the constitutive law. This draws upon fluctuation distribution schemes (upwinding), different combinations of ‘flux’ and ‘median-dual-cell’ spatial discretisations and time-term treatments. Here, unstructured and structured meshes may be used, based largely on triangular grids. A comparison of the two finite volume approaches will be presented, concentrating upon the new features posed by the pom-pom class of models.

Evaluation of High-Order Spectral Volume Method for Benchmark Computational Aeroacoustic Problems

A time-accurate, high-order finite volume method named spectral volume (SV) method has been developed recently for conservation laws on unstructured grids and successfully demonstrated for Euler equations. In this paper, the SV method is evaluated for several benchmark problems in computational aeroacoustics (CAA) to demonstrate its potential for CAA applications. Both one-dimensional and two-dimensional problems are tested in the evaluation. It is shown that the higher-order SV schemes can achieve the same accuracy at a much lower cost than the lower-order ones.

High-order finite volume schemes on unstructured grids using moving least-squares reconstruction. Application to shallow water dynamics

Aceptado para su publicacion en International journal for numerical methods in engineering, el 08/06/2005

Central finite volume methods with constrained transport divergence treatment for ideal MHD

Two and three-dimensional finite volume extensions of the Lax–Friedrichs (LF) and Nessyahu–Tadmor one-dimensional difference schemes were previously presented and successfully applied to several problems for nonlinear hyperbolic systems, and in particular to typical test cases for both inviscid and viscous compressible flows. These “central” schemes by-pass the resolution, at the cell interfaces, of the Riemann problems, thanks to the use of the staggered LF scheme which serves as the base scheme on which high order finite volume methods can be constructed using van Leer’s MUSCL-type limited reconstruction principle. For this purpose, two dual grids are used at alternate time steps. These methods are extended here to several problems in one- and multi-dimensional ideal compressible magnetohydrodynamics using a modified version of the first author’s central methods with oblique (diamond shaped) dual cells. In two-dimensions the system has eight equations and solving the corresponding Riemann problem is an elaborate and time-consuming process. Central methods lead to significant computing time reductions, and the numerical experiments presented here suggest the accuracy is quite satisfactory. In order to satisfy the physical constraint ∇ · B = 0, we have constructed a strategy (“CTCS”) inspired from the Constrained Transport method of Evans and Hawley. The validity of our base scheme and our CTCS approach is clearly confirmed by the results.

Comparison of advection schemes for high-order h–p finite element and finite volume methods

We review and compare advection schemes designed for high-order finite element/finite volume methods. The emphasis is on studying, by numerical examples, the properties of these schemes in terms of accuracy, and monotonicity, and their viability for oceanic applications. The schemes reviewed are classical spectral element, Taylor Galerkin Least Square method, the Discontinuous Galerkin method and high-order finite volume method. The latter two schemes exhibit a definite robustness due to their small, but finite, inherent numerical dissipation. They also prove the most flexible since their discontinuous representation of the solution allows easy implementations of flux limiting or adaptive procedure. Finally, an ad-hoc but simple adaptive procedure is presented to illustrate DGM’s potential; this procedure proved to be extremely effective at controlling Gibbs oscillations in 1D but was too dissipative on the Hecht problem.

Higher order finite volume methods on selfadaptive grids for convection dominated reactive transport problems in porous media

In this paper we study first and higher order finite volume methods for the approximation of reactive systems of transport equations. For the first order scheme an a posteriori error estimate is derived which generalizes the result of [24], as also nonlinear adsorption is taken into account. Based on the a posteriori result, we define strategies for grid adaptation for both methods. Numerical results show the efficiency of the adaptive higher order method, in particular in comparison with first order schemes on uniform computational grids.

High-order finite volume methods for viscoelastic flow problems

In this paper accurate and stable finite volume schemes for solving viscoelastic flow problems are presented. Two contrasting finite volume schemes are described: a hybrid cell-vertex scheme and a pure cell-centred counterpart. Both schemes employ a time-splitting algorithm to evolve the solution through time towards steady state. In the case of the hybrid scheme, a semi-implicit formulation is employed in the momentum equation, based on the Taylor–Galerkin approach with a pressure-correction step to enforce incompressibility. The basis of the pure finite volume approach is a backward Euler scheme with a semi-Lagrangian step to treat the convection terms in the momentum and constitutive equations. Two distinct finite volume schemes are presented for solving the systems of partial differential equations describing the flow of viscoelastic fluids. The schemes are constructed to be second-order accurate in space. The issue of stability is also addressed with respect to the treatment of convection. Numerical examples are presented illustrating the performance of these schemes on some steady and transient problems that possess analytical solutions.

On the application of congruent upwind discretizations for large eddy simulations

Upwind schemes were judged inappropriate for performing accurate large eddy simulations of turbulent flow owing to the artificial dissipation that is present at high wavenumbers of the energy content. Such a conclusion has been drawn also from some results obtained by adopting Finite Difference schemes. The present paper illustrates the performances of some new Finite Volume upwind discretization of the convective terms in the case of the 1-D Burgers model equation while studying the effects of numerical discretization on several Sub-Grid Scales turbulence models, starting from the classical static and dynamic eddy viscosity models through the recent deconvolution-based ones. Basing on previously published papers, large eddy simulations along with a deconvolution-based procedure for de-filtering the evolving variable, have been originally developed and applied. It will be shown how the coherent application of the procedure allows us to develop high-order accurate Finite Volume upwind schemes, which maintain a good spectral resolution in the entire range of resolved scale. Such schemes can be candidate for performing accurate simulations of real turbulence.