Centered Finite Volume Research Articles

A class of central unstaggered schemes for nonlocal conservation laws: Applications to traffic flow models

This study introduces a new class of central unstaggered finite volume methods that are used to approximate solutions to nonlocal conservation laws. The proposed method is based on Nessyahu and Tadmor's (NT). Instead of solving Riemann's problems at the level of cell interfaces, as in the NT scheme, the approach we develop implicitly uses ghost cells while still generating the numerical solution on a single grid. We use our method with the aim of solving one-dimensional nonlocal traffic flow problems. The numerical results we present demonstrate the accuracy, high resolution, and non-oscillatory nature of the proposed method and compare very favorably with those obtained using the original NT method, demonstrating the expected simplicity of a family of unstaggered central schemes and confirming that nonlocal traffic flow models can be treated very efficiently by the suggested method.

A centered limited finite volume approximation of the momentum convection operator for low‐order nonconforming face‐centered discretizations

AbstractWe propose in this article a discretization of the momentum convection operator for fluid flow simulations on quadrangular or generalized hexahedral meshes. The space discretization is performed by the low‐order nonconforming Rannacher–Turek finite element: the scalar unknowns are associated with the cells of the mesh while the velocities unknowns are associated with the edges or faces. The momentum convection operator is of finite volume type, and its expression is derived, as in MUSCL schemes, by a two‐step technique: computation of a tentative flux, here, with a centered approximation of the velocity, and limitation of this flux using monotonicity arguments. The limitation procedure is of algebraic type, in the sense that its does not invoke any slope reconstruction, and is independent from the geometry of the cells. The derived discrete convection operator applies both to constant or variable density flows and may thus be implemented in a scheme for incompressible or compressible flows. To achieve this goal, we derive a discrete analogue of the computation (with the velocity, one of its component, the density, and assuming that the mass balance holds) and discuss two applications of this result: first, we obtain stability results for a semi‐implicit in time scheme for incompressible and barotropic compressible flows; second, we build a consistent, semi‐implicit in time scheme that is based on the discretization of the internal energy balance rather than the total energy. The performance of the proposed discrete convection operator is assessed by numerical tests on the incompressible Navier–Stokes equations, the barotropic and the full compressible Navier–Stokes equations and the compressible Euler equations.

Combining a Central Scheme with the Subtraction Method for Shallow Water Equations

We present a new numerical scheme that is a well-balanced and second-order accurate for systems of shallow water equations (SWEs) with variable bathymetry. We extend in this paper the subtraction method (resulting in well-balancing) to the case of unstaggered central finite volume methods that computes the numerical solution on a single grid. In addition, the proposed scheme avoids solving Riemann problems occurring at cell boundaries as it employs intermediately a layer of ghost-staggered cells. The proposed numerical scheme is then implemented and validated. We successfully manage to solve classical SWE problems from the literature featuring steady states and other equilibria. The results of the study are consistent with previous research, which supports the use of the proposed method to solve SWEs.

An unstructured preconditioned central difference finite volume multiphase Euler solver for computing inviscid cavitating flows over arbitrary two- and three-dimensional geometries

In the present work, a numerical method is adopted and applied for simulating the inviscid cavitating flows around two- and three-dimensional geometries on unstructured meshes. The algorithm uses the preconditioned multiphase Euler equations discretized by a cell-centered central difference finite volume scheme with suitable dissipation terms. The interface capturing method with three transport equation-based cavitation models, namely the Merkle et al., Singhal et al. and Kunz et al. models are employed for the mass transfer between the liquid and vapor phases to be calculated. The simulations of the steady inviscid cavitating flows are performed around different two- and three-dimensional geometries, namely the NACA0012, NACA66(MOD) and two-element NACA4412-4415 hydrofoils, the hemispherical head shape body and the twisted NACA0009 hydrofoil, and the results are obtained over these geometries with the three cavitation models used for different flow conditions. The effects of different numerical parameters on the accuracy of the solution are also examined by a sensitivity study. The present results are compared with those of performed by other researchers which exhibit good agreement. It is indicated that the solution method adopted based on the preconditioned finite volume multiphase Euler flow solver on unstructured meshes is capable of accurately predicting the surface pressure distribution and the cavity shape over the arbitrary two- and three-dimensional geometries.

A modified zonal method to solve coupled conduction-radiation physics within highly porous large scale digitized cellular porous materials

Due to their distinct textures, porous cellular materials have been of interest to many engineering applications. Energy conversion through such materials, especially at high temperatures, is governed by naturally coupled conduction-radiation physics. Therefore, a thorough understanding of the conduction-radiation behavior within these materials is important to properly design and optimize these materials. Several numerical methods developed over the last few decades allow to solve and study the influence of different textural parameters on coupled conductive-radiative heat transfers. These numerical methods require 3D digitized images of the sample of interest obtained either using 3D imagery technique or generated using numerical algorithms. To better represent the complex geometry, the 3D image of the sample requires high spatial resolution, and for a sample which is representative of involved physics might contain hundreds of millions of voxels which is complex to solve and computationally expensive. To cope with this issue, we developed a parallelized discrete scale numerical approach using cell centered Finite Volume Method (FVM) and deterministic ray tracing to solve coupled heat transfer within highly porous large scale complex cellular materials. At the heart of this method rests a decomposition approach based on modified zonal method which significantly reduces the coupling efforts, memory requirements, and computation time. The results of two test cases presented in this study are cross-verified with those presented in literature and computed using Star-CCM+ software. Finally, the method is applied to virtual fibrous sample. These results present the ability of the solver to consider different boundary conditions, at large temperature gradient across boundaries, and to deal with large samples consisting hundreds of million of voxels while providing accurate results. We also present and discuss the sensitivity of other associated parameters on the final results.

Central finite volume schemes for non-local traffic flow models with Arrhenius-type look-ahead rules

We present a central finite volume method and apply it to a new class of nonlocal traffic flow models with an Arrhenius-type look-ahead interaction. These models can be stated as scalar conservation laws with nonlocal fluxes. The suggested scheme is a development of the Nessyah–Tadmor non-oscillatory central scheme. We conduct several numerical experiments in which we carry out the following actions: i) we show the robustness and high resolution of the suggested method; ii) we compare the equations' solutions with local and nonlocal fluxes; iii) we examine how the look-ahead distance affects the numerical solution.

A Time-Continuous Embedding Method for Scalar Hyperbolic Conservation Laws on Manifolds

A time-continuous (tc-)embedding method is first proposed for solving nonlinear scalar hyperbolic conservation laws with discontinuous solutions (shocks and rarefaction waves) on codimension 1, connected, smooth, and closed manifolds (surface PDEs or SPDEs in {mathbb {R}}^2 and {mathbb {R}}^3). The new embedding method improves upon the classical closest point (cp-)embedding method, which requires re-establishments of the constant-along-normal (CAN-)property of the extension function at every time step, in terms of accuracy and efficiency, by incorporating the CAN-property analytically and explicitly in the embedding equation. The tc-embedding SPDEs are solved by the second-order nonlinear central finite volume scheme with a nonlinear minmod slope limiter in space, and the third-order total variation diminished Runge-Kutta scheme in time. An adaptive nonlinear essentially non-oscillatory polynomial interpolation is used to obtain the solution values at the ghost cells. Numerical results in solving the linear wave equation and the Burgers’ equation show that the proposed tc-embedding method has better accuracy, improved resolution, and reduced CPU times than the classical cp-embedding method. The Burgers’ equation, the traffic flow problem, and the Buckley-Leverett equation are solved to demonstrate the robust performance of the tc-embedding method in resolving fine-scale structures efficiently even in the presence of a shock and the essentially non-oscillatory capturing of shocks and rarefaction waves on simple and complex shaped one-dimensional manifolds. Burgers’ equation is also solved on the two-dimensional torus-shaped and spherical-shaped manifolds.

A robust upwind mixed hybrid finite element method for transport in variably saturated porous media

Abstract. The mixed finite element (MFE) method is well adapted for the simulation of fluid flow in heterogeneous porous media. However, when employed for the transport equation, it can generate solutions with strong unphysical oscillations because of the hyperbolic nature of advection. In this work, a robust upwind MFE scheme is proposed to avoid such unphysical oscillations. The new scheme is a combination of the upwind edge/face centered finite volume method with the hybrid formulation of the MFE method. The scheme ensures continuity of both advective and dispersive fluxes between adjacent elements and allows to maintain the time derivative continuous, which permits employment of high-order time integration methods via the method of lines (MOL). Numerical simulations are performed in both saturated and unsaturated porous media to investigate the robustness of the new upwind MFE scheme. Results show that, contrarily to the standard scheme, the upwind MFE method generates stable solutions without under and overshoots. The simulation of contaminant transport into a variably saturated porous medium highlights the robustness of the proposed upwind scheme when combined with the MOL for solving nonlinear problems.

A mass conserving arbitrary Lagrangian–Eulerian formulation for three‐dimensional multiphase fluid flows

AbstractThe arbitrary Lagrangian–Eulerian (ALE) framework presented in [Sahin and Guventurk, An arbitrary Lagrangian–Eulerian framework with exact mass conservation for the numerical simulation of 2D rising bubble problem.Int J Numer Methods Eng. 2017;112:2110–2134] has been extended to simulate three‐dimensional immiscible incompressible multiphase fluid flows. The div‐stable side centered unstructured finite volume formulation is used for the discretization of the incompressible Navier–Stokes equations. A novel kinematic boundary condition that satisfies the local discrete geometric conservation law (DGCL) is implemented to ensure the mass conservation of each species at the machine precision. The pressure field is treated to be discontinuous across the interface with the discontinuous treatment of density and viscosity. Surface tension force is treated as a tangent force and discretized in a semi‐implicit form. Two different approaches for the computation of unit normal vector have been implemented: the least squares biquadratic surface fitting (LSBSF) and the mean weighted by sine and edge length reciprocals (MWSELR). The resulting large system of algebraic equations is solved in a fully coupled manner in order to improve the time step restrictions. As a preconditioner, an approximate matrix factorization similar to that of the projection method is employed and the parallel algebraic multigrid solver BoomerAMG provided by the HYPRE library, which is accessed through the PETSc library, has been utilized for the scaled discrete Laplacian of pressure and the diagonal blocks of mesh deformation equations. The accuracy of the proposed method is initially validated on the static bubble problem, since the surface tension force is highly sensitive to the accurate evaluation of the unit normal vector and the inaccuracies significantly contribute to unphysical velocities, called parasitic currents. The calculations indicate that the parasitic currents can be reduced to machine precision for the MWSELR method. Then, the proposed approach is applied to the single bubble rising in a viscous quiescent liquid for both low and high density ratios. The calculations produce accurate predictions of the bubble shape, center of mass, rise velocity, and so forth. Furthermore, the mass of each species is conserved at machine precision and discontinuous pressure field is obtained in order to avoid errors due to the incompressibility restriction in the vicinity of liquid‐liquid interfaces at large density and viscosity ratios.

A universal centred high-order method based on implicit Taylor series expansion with fast second order evolution of spatial derivatives

In this paper, a centred universal high-order finite volume method for solving hyperbolic balance laws is presented. The scheme belongs to the family of ADER methods where the Generalized Riemann Problems (GRP) is a building block. The solution to these problems is carried through an implicit Taylor series expansion, which allows the scheme to work very well for stiff source terms. The series expansion is high order for the state and requires the evolution in time of spatial derivatives. A Taylor expansion of second order based on a linearization around the resolved state, is proposed for evolving spatial derivatives. A von Neumann stability analysis is carried out to investigate the range of CFL values for which stability and accuracy are balanced. The scheme implements a centred, low dissipation approach for dealing with the advective part of the system which profits from small CFL values. Numerical tests demonstrate that the present scheme can solve, efficiently, hyperbolic balance laws in both conservative and non-conservative form. The scheme is proven to be well-balanced, the C-property, a classical assessment of well-balancing of non-conservative schemes, is numerically demonstrated. An empirical convergence rate assessment shows that the expected theoretical orders of accuracy are achieved up to the fifth order.

Deep learning surrogate interacting Markov chain Monte Carlo based full wave inversion scheme for properties of materials quantification

Full Wave Inversion (FWI) imaging scheme has many applications in engineering, geoscience and medical sciences. In this paper, a surrogate deep learning FWI approach is presented to quantify properties of materials using stress waves. Such inverse problems, in general, are ill-posed and nonconvex, especially in cases where the solutions exhibit shocks, heterogeneity and discontinuities. The proposed approach is proven efficient to obtain global minima responses in these cases. This approach is trained based on random sampled sets of material properties and sampled trials around local minima, therefore, it requires a forward simulation can handle high heterogeneity, discontinuities and large gradients. High resolution Kurganov–Tadmor (KT) central finite volume method is used as forward wave propagation operator. Using the proposed framework, material properties of 2D media are quantified for several different situations. The results demonstrate the feasibility of the proposed method for estimating mechanical properties of materials with high accuracy using deep learning approaches.

Accuracy Verification of a 2D Adaptive Mesh Refinement Method Using Backward-Facing Step Flow of Low Reynolds Numbers

Identifying centers of vortices of fluid flow accurately is one of the accuracy measures for computational methods. After verifying the accuracy of the 2D adaptive mesh refinement (AMR) method in the benchmarks of 2D lid-driven cavity flow, this paper shows the accuracy verification by the benchmarks of 2D backward-facing step flow. The AMR method refines a mesh using the numerical solution of the Navier–Stokes equations computed on the mesh by an open source software Navier2D which implemented a vertex centered finite volume method (FVM) using the median dual mesh to form control volumes about each vertex. The accuracy is shown by the comparison between vortex center locations calculated from the linearly interpolated numerical solutions and those obtained in the benchmark. The AMR method is proposed based on the qualitative theory of differential equations, and it can be applied to refine a mesh as many times as required and used to seek accurate numerical solutions of the mathematical models including the continuity equation for incompressible fluid or steady-state compressible flow with low computational cost.

Well-balanced central schemes for the one and two-dimensional Euler systems with gravity

In this paper we develop a family of central schemes for the one and two-dimensional systems of Euler equations with gravitational source term. The proposed schemes are unstaggered, second-order, central finite volume schemes that avoid solving Riemann problems at the cell interfaces and avoid switching between an original and a staggered grid. The main feature of the schemes developed here is that they are capable of preserving any steady state of the Euler with gravity system up to machine accuracy by updating the numerical solution in terms of a relevant reference solution. The methodology proposed results in a well-balanced scheme capable of capturing any steady state. Our scheme is then implemented and used to solve classical problems from the recent literature.

Modifications to the gradient schemes on unstructured cell centered grids for the accurate determination of gradients near conductivity changes

Gradient schemes for the cell centered finite volume method on unstructured grids, namely, the divergence theorem and the least squares schemes, have been widely adopted because they have reached a high precision for most applications. These schemes assume continuously differentiable fields for the calculation of the gradients. However, this assumption is violated in the vicinity of conductivity jumps between cells. It is shown that this deficiency leads to a wrong calculation of the gradients and thus the flux density in cells near conductivity changes. For large conductivity jumps, the error of the flux density can exceed several orders of magnitude. Based on theoretical considerations, flux conservative versions of the schemes are derived for the central gradient scheme and extended to the divergence theorem and least squares schemes. The modified schemes named flux conservative divergence theorem and flux conservative least squares take the nonlinearity of a conservation variable near conductivity changes into account and eliminate the error made by the assumption of a continuously differentiable field. The schemes are demonstrated on Cartesian and highly skewed grids with different grid resolutions with a large conductivity jump. The error of the flux density is shown to be reduced by several orders of magnitude up to machine precision for Cartesian grids.

A Total Lagrangian upwind Smooth Particle Hydrodynamics algorithm for large strain explicit solid dynamics

In previous work (Lee et al., 2016, 2017), Lee et al. introduced a new Smooth Particle Hydrodynamics (SPH) computational framework for large strain explicit solid dynamics with special emphasis on the treatment of near incompressibility. A first order system of hyperbolic equations was presented expressed in terms of the linear momentum and the minors of the deformation, namely the deformation gradient, its co-factor and its Jacobian. Taking advantage of this representation, the suppression of numerical deficiencies (e.g. spurious pressure, long term instability and/or consistency issues) was addressed through well-established stabilisation procedures. In Lee et al. (2016), the adaptation of the very efficient Jameson–Schmidt–Turkel algorithm was presented. Lee et al. (2017) introduced an adapted variationally consistent Streamline Upwind Petrov–Galerkin methodology. In this paper, we now introduce a third alternative stabilisation strategy, extremely competitive, and which does not require the selection of any user-defined artificial stabilisation parameter. Specifically, a characteristic-based Riemann solver in conjunction with a linear reconstruction procedure is used, with the aim to guarantee both consistency and conservation of the overall algorithm. We show that the proposed SPH formulation is very similar in nature to that of the upwind vertex centred Finite Volume Method presented in Aguirre et al. (2015). In order to extend the application range towards the incompressibility limit, an artificial compressibility algorithm is also developed. Finally, an extensive set of challenging numerical examples is analysed. The new SPH algorithm shows excellent behaviour in compressible, nearly incompressible and truly incompressible scenarios, yielding second order of convergence for velocities, deviatoric and volumetric components of the stress.

Grid Average and Nodal Value in High Order Scheme

ABSTRACTThe concepts of nodal value and grid average in cell centered finite volume method (FVM) are clarified in this work, strict distinction between the two concepts in constructing numerical schemes is made, and common fault in misidentifying the two concepts is pointed out. The expansion based on grid average, similar to Taylor’s expansion, is deduced to construct correct scheme in terms of grid average and to obtain modified partial differential equation (MPDE) which determines the order of accuracy of numerical scheme theoretically. Correct high order scheme, taking QUICK (Quadratic Upstream Interpolation for Convective Kinematics) scheme as an example, is constructed in different approaches. Furthermore, the property of interpolation coefficients is analyzed. We also pointed out that for high order schemes, round-off error dominates the absolute error in fine grid and truncation error dominates the absolute error in coarse grid.

An upwind cell centred Total Lagrangian finite volume algorithm for nearly incompressible explicit fast solid dynamic applications

The paper presents a new computational framework for the numerical simulation of fast large strain solid dynamics, with particular emphasis on the treatment of near incompressibility. A complete set of first order hyperbolic conservation equations expressed in terms of the linear momentum and the minors of the deformation (namely the deformation gradient, its co-factor and its Jacobian), in conjunction with a polyconvex nearly incompressible constitutive law, is presented. Taking advantage of this elegant formalism, alternative implementations in terms of entropy-conjugate variables are also possible, through suitable symmetrisation of the original system of conservation variables. From the spatial discretisation standpoint, modern Computational Fluid Dynamics code “OpenFOAM” [http://www.openfoam.com/] is here adapted to the field of solid mechanics, with the aim to bridge the gap between computational fluid and solid dynamics. A cell centred finite volume algorithm is employed and suitably adapted. Naturally, discontinuity of the conservation variables across control volume interfaces leads to a Riemann problem, whose resolution requires special attention when attempting to model materials with predominant nearly incompressible behaviour (κ∕μ≥500). For this reason, an acoustic Riemann solver combined with a preconditioning procedure is introduced. In addition, a global a posteriori angular momentum projection procedure proposed in Haider et al. (2017) is also presented and adapted to a Total Lagrangian version of the nodal scheme of Kluth and Després (2010) used in this paper for comparison purposes. Finally, a series of challenging numerical examples is examined in order to assess the robustness and applicability of the proposed methodology with an eye on large scale simulation in future works.

Well-balanced central schemes for two-dimensional systems of shallow water equations with wet and dry states

• New central finite volume scheme for the two-dimensional system of shallow water equations was developed. • The developed scheme is well-balanced and preserves the lake at rest constraint. • The developed scheme is capable of handling wet and dry states whenever water run-ups/drains in the computational domain. • The developed scheme is validated and used to solve classical two-dimensional shallow water equation problems. The aim of this paper is to develop a new second-order accurate central scheme for the numerical solution of the two-dimensional system of shallow water equations (SWE) featuring wet and dry states over variable waterbeds. The proposed central scheme follows a classical Riemann-free finite volume method and evolves the numerical solution of systems of hyperbolic balance laws on a single Cartesian grid. Furthermore, the proposed well-balanced scheme preserves the lake at rest constraint thanks to a careful well-balanced discretization of the SWE system, and allows a proper interaction between wet and dry states whenever water run-ups/drains arise. For verification purposes, classical SWE problems appearing in the recent literature are successfully solved.

Weak Galerkin Finite Element Methods for the Simulation of Single-Phase Flow in Fractured Porous Media

This paper presents a numerical simulation of the flow in fractured porous media, which can be efficiently described by the reduced model consisting of the bulk problem in the porous matrix and the fracture problem in the fracture together with physically consistent coupling conditions. The reduced model is solved by using two types of weak Galerkin finite element methods (on simplex mesh and polygonal mesh, respectively) for the bulk problem coupled with the cell centered finite volume method for the fracture problem. We prove that the algebraic system arising in the discrete formulation is symmetric and positive-definite, which leads to the well-posedness of the discrete problem. Error estimates for the pressure in suitable norms are established. A series of numerical experiments demonstrate the accuracy and robustness of our method with respect to the shape of the grid and the permeability of the fractures.

Numerical Study on Mixed Convection in Ventilated Cavities with Different Aspect Ratios

An unsteady numerical investigation on mixed convection in a two dimensional open ended cavity with different aspect ratios is carried out. In this investigation, uniform temperature is set to the left and the right sides of the cavity while the other surfaces are adiabatic. The simulation is performed for a wide range of Reynolds numbers (Re = 100–1000) and Richardson numbers (Ri = 0.132–6.5 × 102), and various cavity aspect ratios (L/D = 0.5–4.0) and H/D = 0.1. Governing equations are solved using a cell centered finite volume code, a SIMPLE numerical projection scheme and a 2nd order accuracy. Results are presented in the form of streamlines, isothermal lines, and velocity profiles in the channel. The conclusion is that the enhancement of heat transfer rate is generated principally by the increasing Re and the assisting configuration is thermally more efficient when compared to the opposing one.