Centered Finite Volume Method 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.

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.

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 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.

Mass transport in a moist planetary climate model

Planetary climate models (PCMs) are developed to explore planetary climates other than that of the Earth. Therefore, the methods implemented need to be suitable for a large diversity of conditions. Every planet with a significant atmosphere has condensible cycles (e.g. the hydrological cycle), which can play an essential role in the planet’s appearance and environment. We must accurately represent a condensible cycle in our planet simulations to build a powerful planetary climate predictor. OASIS is a 3D PCM capable of self-consistently representing the main physical processes that drive a planet’s environment. In this work, we improve the representation of mass transport in OASIS, which is the first step towards a complete and flexible implementation of a condensible cycle. We implement an upwind-biased scheme on a piece-wise linear approximation with a flux limiter to solve the mass transport equation. We first benchmark the new scheme on a 2D problem that confirms the superior properties of the new method over the central finite-volume method in terms of performance, accuracy, and shape-preserving mass distribution. Due to the new scheme’s less dispersive nature, we do not have to apply any unphysical diffusion to maintain the model stable. OASIS includes the new improved solver in the total mass and the tracer (e.g. clouds and individual gas chemical species) transport. We couple the new formulation with physical schemes and validate the new code on two 3D simulations of an ocean Earth-like planet and an ocean tidally locked planet. The new OASIS simulations are robust and do not show any known problems from the dynamics-physics coupling. We show that the two simulations capture the main characteristics of ocean planet atmospheres and are easy to set up. We propose these two simulations as the first standard benchmark tests for models built to explore moist planetary environments.

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.

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.

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.

Locally Conservative Continuous Galerkin FEM for Pressure Equation in Two-Phase Flow Model in Subsurfaces

A typical two-phase model for subsurface flow couples the Darcy equation for pressure and a transport equation for saturation in a nonlinear manner. In this paper, we study a combined method consisting of continuous Galerkin finite element methods (CGFEMs) followed by a post-processing technique for Darcy equation and a nodal centered finite volume method (FVM) with upwind schemes for the saturation transport equation, in which the coupled nonlinear problem is solved in the framework of operator decomposition. The post-processing technique is applied to CGFEM solutions to obtain locally conservative fluxes which ensures accuracy and robustness of the FVM solver for the saturation transport equation. We applied both upwind scheme and upwind scheme with slope limiter for FVM on triangular meshes in order to eliminate the non-physical oscillations. Various numerical examples are presented to demonstrate the performance of the overall methodology.

Generalized SIMPLE-Based Pressure Correction Method for Unstructured Colocated Grids

This paper reports on the SIMPLE-like solution algorithm that significantly improves velocity–pressure coupling, leading to the accelerated convergence. The algorithm is applicable to both incompressible and compressible flows. It is implemented within the framework of a cell centered finite volume method using colocated storage of flow variables on unstructured grids. The convergence acceleration is an outcome of the novel treatment of the pressure correction equation that accounts for two commonly neglected terms appearing in a general pressure correction equation. These terms, namely, the neighbor and nonorthogonal corrections, represent velocity corrections from neighboring cells and the cell-face pressure correction gradient associated with the grid nonorthogonality, respectively. Reminiscent of the PISO method, both neighbor and nonorthogonal corrections are taken into account by performing two or more correction steps. However, the full inclusion of neighbor velocity corrections can prevent the solution convergence. Following an analogy between time marching and a steady-state iterative approach, this problem has been resolved by introducing a pseudounsteady term into discretized velocity correction equations. Using this term, the contribution of neighbor corrections is underrelaxed, enabling a satisfactory convergence rate. The algorithm is applied to several benchmark test cases, covering laminar and turbulent, as well as incompressible and compressible, flows. For all test cases, the convergence rate can be significantly improved by performing two or more pressure correction steps. An optimal number of pressure corrections exists for which a meaningful reduction of computing time is possible.

Well-balanced central finite volume methods for the Ripa system

We propose a new well-balanced central finite volume scheme for the Ripa system both in one and two space dimensions. The Ripa system is a nonhomogeneous hyperbolic system with a non-zero source term that is obtained from the shallow water equations system by incorporating horizontal temperature gradients. The proposed numerical scheme is a second-order accurate finite volume method that evolves a non-oscillatory numerical solution on a single grid, avoids the process of solving Riemann problems arising at the cell interfaces, and follows a well-balanced discretization that ensures the steady state requirement by discretizing the geometrical source term according to the discretization of the flux terms. Furthermore the proposed scheme mimics the surface gradient method and discretizes the water height according to the discretization of the water level. The proposed scheme is then applied and classical one and two-dimensional Ripa problems with flat or variable bottom topographies are successfully solved. The obtained numerical results are in good agreement with corresponding ones appearing in the recent literature, thus confirming the potential and efficiency of the proposed method.

High-Order Finite-Volume Transport on the Cubed Sphere: Comparison between 1D and 2D Reconstruction Schemes

Abstract This paper presents two finite-volume (FV) schemes for solving linear transport problems on the cubed-sphere grid system. The schemes are based on the central-upwind finite-volume (CUFV) method, which is a class of Godunov-type method for solving hyperbolic conservation laws, and combines the attractive features of the classical upwind and central FV methods. One of the CUFV schemes is based on a dimension-by-dimension approach and employs a fifth-order one-dimensional (1D) Weighted Essentially Nonoscillatory (WENO5) reconstruction method. The other scheme employs a fully two-dimensional (2D) fourth-order accurate reconstruction method. The cubed-sphere grid system imposes several computational challenges due to its patched-domain topology and nonorthogonal curvilinear grid structure. A high-order 1D interpolation procedure combining cubic and quadratic interpolations is developed for the FV schemes to handle the discontinuous edges of the cubed-sphere grid. The WENO5 scheme is compared against the fourth-order Kurganov–Levy (KL) scheme formulated in the CUFV framework. The performance of the schemes is compared using several benchmark problems such as the solid-body rotation and deformational-flow tests, and empirical convergence rates are reported. In addition, a bound-preserving filter combined with an optional positivity-preserving filter is tested for nonsmooth problems. The filtering techniques considered are local, inexpensive, and effective. A fourth-order strong stability preserving explicit Runge–Kutta time-stepping scheme is used for integration. The results show that schemes are competitive to other published FV schemes in the same category.

A novel finite volume method for cylindrical heat conduction problems

A new finite volume method for cylindrical heat conduction problems based on Lnr-type diffusion equation is proposed in this paper with detailed derivation. On the basis of coordinate transformation, the diffusion term in the r direction of the heat conduction equation in a cylindrical coordinate is transformed into the Lnr-type diffusion term. Considering the influence of different boundary conditions, source terms and ratios of the internal to external radius, four typical categories of cases are calculated by the newly proposed method based on Lnr-type diffusion equation, the method based on local analytical solution and traditional central difference finite volume method respectively. The comparison of calculation results indicates that the new finite volume method is more accurate than the conventional one, since when the governing equation is discretized, the new method transforms λr∂T∂r into λ∂T∂lnr and treats this term as a whole to guarantee the conservativeness of diffusion flux. Numerical results also show that the total calculating time and required computation grid number of the new method are significantly less than the other two methods for achieving the same level of accuracy.

Central Finite Volume Methods Applied in Relativistic Magnetohydrodynamics: Applications in Disks and Jets

We have developed a new computer program in Fortran 90, in order to obtain numerical solutions of a system of Relativistic Magnetohydrodynamics partial differential equations with predetermined gravitation (GRMHD), capable of simulating the formation of relativistic jets from the accretion disk of matter up to his ejection. Initially we carried out a study on numerical methods of unidimensional Finite Volume, namely Lax-Friedrichs, Lax-Wendroff, Nessyahu-Tadmor method and Godunov methods dependent on Riemann problems, applied to equations Euler in order to verify their main features and make comparisons among those methods. It was then implemented the method of Finite Volume Centered of Nessyahu-Tadmor, a numerical schemes that has a formulation free and without dimensional separation of Riemann problem solvers, even in two or more spatial dimensions, at this point, already applied in equations GRMHD. Finally, the Nessyahu-Tadmor method was possible to obtain stable numerical solutions without spurious oscillations or excessive dissipation from the magnetized accretion disk process in rotation with respect to a central black hole (BH) Schwarzschild and immersed in a magnetosphere, for the ejection of matter in the form of jet over a distance of fourteen times the radius of the BH, a record in terms of astrophysical simulation of this kind. Also in our simulations, we managed to get substructures jets. A great advantage obtained was that, with the our code, we got simulate GRMHD equations in a simple personal computer. Keywords—Finite Volume Methods, Central Schemes, Fortran 90, Relativistic Astrophysics, Jet.

Numerical analysis of unsteady cavitating flow in an axial inducer

This work presents the results of numerical simulation of unsteady cavitating flow through a two–bladed axial inducer. First, the analysis was carried out in a blade cascade, this two–dimensional simplified model, obtained from the studied axial inducer, was used as a test case. Later, the numerical simulations were extended to the original three-dimensional inducer. All numerical calculations were realized in cavitating flow regime. Initially, the results were obtained in steady state, and then in unsteady state.The main purpose of this study is to explore the local cavitation instabilities, such as alternate blade cavitation and rotating blade cavitation, which can appear in this type of devices when they work under certain operating conditions.The numerical results show that the fluid flow in the axial inducer is altered by the emergence of the cavitation. These vapor regions are formed, firstly near to the leading edge of each blade. The behavior of the cavitation depends on the operating conditions of the inducer, mainly by the flow rate and the suction pressure.The numerical simulation was performed using a commercial code based on a cell–centered finite–volume method. The cavitation model used for calculations assumes a thermal equilibrium between phases. It is based on the classical conservation equations of the vapor phase and a mixture phase, with mass transfer due to the cavitation appearing as a source and a sink term in the vapor mass fraction equation. The mass transfer rate is derived from a simplified Rayleigh–Plesset model for bubble dynamics.