High Order Finite Volume Schemes Research Articles

Accuracy of numerical simulations of contaminant transport in heterogeneous aquifers: A comparative study

This work deals with a comparison of different numerical schemes for the simulation of contaminant transport in heterogeneous porous media. The numerical methods under consideration are Galerkin finite element (GFE), finite volume (FV), and mixed hybrid finite element (MHFE). Concerning the GFE we use linear and quadratic finite elements with and without upwind stabilization. Besides the classical MHFE a new and an upwind scheme are tested. We consider higher order finite volume schemes as well as two time discretization methods: backward Euler (BE) and the second order backward differentiation formula BDF (2). It is well known that numerical (or artificial) diffusion may cause large errors. Moreover, when the Péclet number is large, a numerical code without some stabilising techniques produces oscillating solutions. Upwind schemes increase the stability but show more numerical diffusion. In this paper we quantify the numerical diffusion for the different discretization schemes and its dependency on the Péclet number. We consider an academic example and a realistic simulation of solute transport in heterogeneous aquifer. In the latter case, the stochastic estimates used as reference were obtained with global random walk (GRW) simulations, free of numerical diffusion. The results presented can be used by researchers to test their numerical schemes and stabilization techniques for simulation of contaminant transport in groundwater.

High order finite volume methods on wavelet-adapted grids with local time-stepping on multicore architectures for the simulation of shock-bubble interactions

We present a space–time adaptive solver for single- and multi-phase compressible flows that couples average interpolating wavelets with high-order finite volume schemes. The solver introduces the concept of wavelet blocks, handles large jumps in resolution and employs local time-stepping for efficient time integration. We demonstrate that the inherently sequential wavelet-based adaptivity can be implemented efficiently in multicore computer architectures using task-based parallelism and introducing the concept of wavelet blocks. We validate our computational method on a number of benchmark problems and we present simulations of shock-bubble interaction at different Mach numbers, demonstrating the accuracy and computational performance of the method.

On the benefits of using GPUS to simulate shallow flows with finite volume schemes

In this paper, we focus on the efficient implementation of path conservative Roe type high order finite volume schemes to simulate shallow flows. The motion of a layer of homogeneous non-viscous fluid is supposed to be governed by the shallow-water system, formulated under the form of a conservation law with source terms. The implementation of the scheme is carried out on Graphics Processing Units (GPUs), thus achieving a substantial improvement of the speedup with respect to normal CPUs. Finally, some numerical experiments are presented.

On the simulation of wave propagation with a higher-order finite volume scheme based on Reproducing Kernel Methods

In this work we study the dispersion and dissipation characteristics of a higher-order finite volume method based on Moving Least Squares approximations (FV-MLS), and we analyze the influence of the kernel parameters on the properties of the scheme. Several numerical examples are included. The results clearly show a significant improvement of dispersion and dissipation properties of the numerical method if the third-order FV-MLS scheme is used compared with the second-order one. Moreover, with the explicit fourth-order Runge–Kutta scheme the dispersion error is lower than with the third-order Runge–Kutta scheme, whereas the dissipation error is similar for both time-integration schemes. It is also shown than a CFL number lower than 0.8 is required to avoid an unacceptable dispersion error.

Resolution of computational aeroacoustics problems on unstructured grids with a higher-order finite volume scheme

Computational fluid dynamics (CFD) has become increasingly used in the industry for the simulation of flows. Nevertheless, the complex configurations of real engineering problems make the application of very accurate methods that only work on structured grids difficult. From this point of view, the development of higher-order methods for unstructured grids is desirable. The finite volume method can be used with unstructured grids, but unfortunately it is difficult to achieve an order of accuracy higher than two, and the common approach is a simple extension of the one-dimensional case. The increase of the order of accuracy in finite volume methods on general unstructured grids has been limited due to the difficulty in the evaluation of field derivatives. This problem is overcome with the application of the Moving Least Squares (MLS) technique on a finite volume framework. In this work we present the application of this method (FV-MLS) to the solution of aeroacoustic problems.

ADER schemes on unstructured meshes for nonconservative hyperbolic systems: Applications to geophysical flows

We develop a new family of well-balanced path-conservative quadrature-free one-step ADER finite volume and discontinuous Galerkin finite element schemes on unstructured meshes for the solution of hyperbolic partial differential equations with non-conservative products and stiff source terms. The fully discrete formulation is derived using the recently developed framework of explicit one-step P N P M schemes of arbitrary high order of accuracy in space and time for conservative hyperbolic systems [Dumbser M, Balsara D, Toro EF, Munz CD. A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes. J Comput Phys 2008;227:8209–53]. The two key ingredients of our high order approach are: first, the high order accurate P N P M reconstruction operator on unstructured meshes, using the WENO strategy presented in [Dumbser M, Käser M, Titarev VA Toro EF. Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. J Comput Phys 2007;226:204–43] to ensure monotonicity at discontinuities, and second, a local space–time Galerkin scheme to predict the evolution of the reconstructed polynomial data inside each element during one time step to obtain a high order accurate one-step time discretization. This approach is also able to deal with stiff source terms as shown in [Dumbser M, Enaux C, Toro EF. Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. J Comput Phys 2008;227:3971–4001]. These two key ingredients are combined with the recently developed path-conservative methods of Parés [Parés C. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J Numer Anal 2006;44:300–21] and Castro et al. [Castro MJ, Gallardo JM, Parés C. High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems. Math Comput 2006;75:1103–34] to treat the non-conservative products properly. We show applications of our method to the two-layer shallow water equations as well as applications to the recently published depth-averaged two-fluid flow model of Pitman and Le [Pitman EB, Le L. A two-fluid model for avalanche and debris flows. Philos Trans Roy Soc A 2005;363:1573–601].

Characteristic-based finite volume scheme for 1D Euler equations

In this paper, a high-order finite-volume scheme is presented for the onedimensional scalar and inviscid Euler conservation laws. The Simpson’s quadrature rule is used to achieve high-order accuracy in time. To get the point value of the Simpson,s quadrature, the characteristic theory is used to obtain the positions of the grid points at each sub-time stage along the characteristic curves, and the third-order and fifth-order central weighted essentially non-oscillatory (CWENO) reconstruction is adopted to estimate the cell point values. Several standard one-dimensional examples are used to verify the high-order accuracy, convergence and capability of capturing shock.

Well-balanced high-order centred schemes for non-conservative hyperbolic systems. Applications to shallow water equations with fixed and mobile bed

This paper concerns the development of high-order accurate centred schemes for the numerical solution of one-dimensional hyperbolic systems containing non-conservative products and source terms. Combining the PRICE-T method developed in [Toro E, Siviglia A. PRICE: primitive centred schemes for hyperbolic system of equations. Int J Numer Methods Fluids 2003;42:1263–91] with the theoretical insights gained by the recently developed path-conservative schemes [Castro M, Gallardo J, Parés C. High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products applications to shallow-water systems. Math Comput 2006;75:1103–34; Parés C. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J Numer Anal 2006;44:300–21], we propose the new PRICE-C scheme that automatically reduces to a modified conservative FORCE scheme if the underlying PDE system is a conservation law. The resulting first-order accurate centred method is then extended to high order of accuracy in space and time via the ADER approach together with a WENO reconstruction technique. The well-balanced properties of the PRICE-C method are investigated for the shallow water equations. Finally, we apply the new scheme to the shallow water equations with fix bottom topography and with variable bottom solving an additional sediment transport equation.

A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes

In this article, a conservative least-squares polynomial reconstruction operator is applied to the discontinuous Galerkin method. In a first instance, piecewise polynomials of degree N are used as test functions as well as to represent the data in each element at the beginning of a time step. The time evolution of these data and the flux computation, however, are then done with a different set of piecewise polynomials of degree M ⩾ N , which are reconstructed from the underlying polynomials of degree N. This approach yields a general, unified framework that contains as two special cases classical high order finite volume (FV) schemes ( N = 0 ) as well as the usual discontinuous Galerkin (DG) method ( N = M ) . In the first case, the polynomial is reconstructed from cell averages, for the latter, the reconstruction reduces to the identity operator. A completely new class of numerical schemes is generated by choosing N ≠ 0 and M > N . The reconstruction operator is implemented for arbitrary polynomial degrees N and M on unstructured triangular and tetrahedral meshes in two and three space dimensions. To provide a high order accurate one-step time integration of the same formal order of accuracy as the spatial discretization operator, the (reconstructed) polynomial data of degree M are evolved in time locally inside each element using a new local continuous space–time Galerkin method. As a result of this approach, we obtain, as a high order accurate predictor, space–time polynomials for the vector of conserved variables and for the physical fluxes and source terms, which then can be used in a natural way to construct very efficient fully-discrete and quadrature-free one-step schemes. This feature is particularly important for DG schemes in three space dimensions, where the cost of numerical quadrature may become prohibitively expensive for very high orders of accuracy. Numerical convergence studies of all members of the new general class of proposed schemes are shown up to sixth-order of accuracy in space and time on unstructured two- and three-dimensional meshes for two very prominent nonlinear hyperbolic systems, namely for the Euler equations of compressible gas dynamics and the equations of ideal magnetohydrodynamics (MHD). The results indicate that the new class of intermediate schemes ( N ≠ 0 , M > N ) is computationally more efficient than classical finite volume or DG schemes. Finally, a large set of interesting test cases is solved on unstructured meshes, where the proposed new time stepping approach is applied to the equations of ideal and relativistic MHD as well as to nonlinear elasticity, using a standard high order WENO finite volume discretization in space to cope with discontinuous solutions.

High-order Finite Volume Methods and Multiresolution Reproducing Kernels

This paper presents a review of some of the most successful higher-order numerical schemes for the compressible Navier-Stokes equations on unstructured grids. A suitable candidate scheme would need to be able to handle potentially discontinuous flows, arising from the predominantly hyperbolic character of the equations, and at the same time be well suited for elliptic problems, in order to deal with the viscous terms. Within this context, we explore the performance of Moving Least-Squares (MLS) approximations in the construction of higher order finite volume schemes on unstructured grids. The scope of the application of MLS is threefold: 1) computation of high order derivatives of the field variables for a Godunov-type approach to hyperbolic problems or terms of hyperbolic character, 2) direct reconstruction of the fluxes at cell edges, for elliptic problems or terms of elliptic character, and 3) multiresolution shock detection and selective limiting. The proposed finite volume method is formulated within a continuous spatial representation framework, provided by the MLS approximants, which is “broken” locally (inside each cell) into piecewise polynomial expansions, in order to make use of the specialized finite volume technology for hyperbolic problems. This approach is in contrast with the usual practice in the finite volume literature, which proceeds bottom-up, starting from a piecewise constant spatial representation. Accuracy tests show that the proposed method achieves the expected convergence rates. Representative simulations show that the methodology is applicable to problems of engineering interest, and very competitive when compared to other existing procedures.

A positive and conservative second order finite volume scheme applied to a phosphor cycle in canals with sediment

AbstractThis paper is devoted to the description of a higher order finite volume scheme for the 2D shallow water equation with a specific focus on a complex phosphor cycle describing processes in lakes and rivers. The scheme guarantees the unconditional positivity and mass conservation of respectively modelled quantities, which are crucial properties for meaningful computed results in biochemical contexts. It works on secondary unstructured meshes which enables the application to arbitrary complex geometries. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Arbitrary high-order finite volume schemes for seismic wave propagation on unstructured meshes in 2D and 3D

SUMMARY We present a new numerical method to solve the heterogeneous anelastic seismic wave equations with arbitrary high order of accuracy in space and time on unstructured triangular and tetrahedral meshes in two and three space dimensions, respectively. Using the velocity–stress formulation provides a linear hyperbolic system of equations with source terms that is completed by additional equations for the anelastic functions including the strain history of the material. These additional equations result from the rheological model of the generalized Maxwell body and permit the incorporation of realistic attenuation properties of viscoelastic material accounting for the behaviour of elastic solids and viscous fluids. The proposed method relies on the finite volume (FV) approach where cell-averaged quantities are evolved in time by computing numerical fluxes at the element interfaces. The basic ingredient of the numerical flux function is the solution of Generalized Riemann Problems at the element interfaces according to the arbitrary high-order derivatives (ADER) approach of Toro et al., where the initial data is piecewise polynomial instead of piecewise constant as it was in the original first-order FV scheme developed by Godunov. The ADER approach automatically produces a scheme of uniformly high order of accuracy in space and time. The high-order polynomials in space, needed as input for the numerical flux function, are obtained using a reconstruction operator acting on the cell averages. This reconstruction operator uses some techniques originally developed in the Discontinuous Galerkin (DG) Finite Element framework, namely hierarchical orthogonal basis functions in a reference element. In particular, in this article we pay special attention to underline the differences as well as the points in common with the ADER-DG schemes previously developed by the authors, especially concerning the MPI parallelization of both methods. The numerical convergence analysis demonstrates that the proposed FV schemes provide very high order of accuracy even on unstructured tetrahedral meshes while computational cost for a desired accuracy can be reduced when applying higher order reconstructions. Applications to a series of well-acknowledged elastic and anelastic test cases and comparisons with analytic and numerical reference solutions, obtained by different well-established numerical methods, confirm the performance of the proposed method. Therefore, the development of the highly accurate ADER-FV approach for tetrahedral meshes including viscoelastic material provides a novel, flexible and efficient numerical technique to approach 3-D wave propagation problems including realistic attenuation and complex geometry.

On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas

We present a finite volume scheme for solving shallow water equations with source term due to the bottom topography. The scheme has the following properties: it is high-order accurate in smooth wet regions, it correctly solves situations where dry areas are present, and it is well-balanced. The scheme is developed within a general nonconservative framework, and it is based on hyperbolic reconstructions of states. The treatment of wet/dry fronts is carried out by solving specific nonlinear Riemann problems at the corresponding intercells.

Finite volume solvers and Moving Least-Squares approximations for the compressible Navier–Stokes equations on unstructured grids

This paper explores the approximation power of Moving Least-Squares (MLS) approximations in the context of higher-order finite volume schemes on unstructured grids. The scope of the application of MLS is threefold: (1) computation of high-order derivatives of the field variables for a Godunov-type approach to hyperbolic problems or terms of hyperbolic character, (2) direct reconstruction of the fluxes at cell edges, for elliptic problems or terms of elliptic character, and (3) multiresolution shock detection and selective limiting. A major advantage of the proposed methodology over the most popular existing higher-order methods is related to the viscous discretization. The use of MLS approximations allows the direct reconstruction of high-order viscous fluxes using quite compact stencils, and without introducing new degrees of freedom, which results in a significant reduction in storage and workload. A selective limiting procedure is proposed, based on the multiresolution properties of the MLS approximants, which allows to switch off the limiters in smooth regions of the flow. Accuracy tests show that the proposed method achieves the expected convergence rates. Representative simulations show that the methodology is applicable to problems of engineering interest.

Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems

In this article we present a quadrature-free essentially non-oscillatory finite volume scheme of arbitrary high order of accuracy both in space and time for solving nonlinear hyperbolic systems on unstructured meshes in two and three space dimensions. For high order spatial discretization, a WENO reconstruction technique provides the reconstruction polynomials in terms of a hierarchical orthogonal polynomial basis over a reference element. The Cauchy–Kovalewski procedure applied to the reconstructed data yields for each element a space–time Taylor series for the evolution of the state and the physical fluxes. This Taylor series is then inserted into a special numerical flux across the element interfaces and is subsequently integrated analytically in space and time. Thus, the Cauchy–Kovalewski procedure provides a natural, direct and cost-efficient way to obtain a quadrature-free formulation, avoiding the expensive numerical quadrature arising usually for high order finite volume schemes in three space dimensions. We show numerical convergence results up to sixth order of accuracy in space and time for the compressible Euler equations on triangular and tetrahedral meshes in two and three space dimensions. Furthermore, various two- and three-dimensional test problems with smooth and discontinuous solutions are computed to validate the approach and to underline the non-oscillatory shock-capturing properties of the method.

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.

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.

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.

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.