Hybrid Finite Volume Research Articles

Comparison of finite-volume schemes for diffusion problems

We present an abstract discretization framework and demonstrate that various cell-centered and hybrid finite-volume schemes fit into it. The different schemes considered in this work are then analyzed numerically for an elliptic model problem with respect to the properties consistency, coercivity, extremum principles, and sparsity. The test cases presented comprise of two- and three-dimensional setups, mildly and highly anisotropic tensors and grids of different complexities. The results show that all schemes show a similar convergence behavior, except for the two-point flux approximation scheme, and seem to be coercive. Furthermore, they confirm that linear schemes, in contrast to nonlinear schemes, are in general neither positivity-preserving nor satisfy discrete minimum or maximum principles.

A hybrid finite volume – finite element method for bulk–surface coupled problems

The paper develops a hybrid method for solving a system of advection–diffusion equations in a bulk domain coupled to advection–diffusion equations on an embedded surface. A monotone nonlinear finite volume method for equations posed in the bulk is combined with a trace finite element method for equations posed on the surface. In our approach, the surface is not fitted by the mesh and is allowed to cut through the background mesh in an arbitrary way. Moreover, a triangulation of the surface into regular shaped elements is not required. The background mesh is an octree grid with cubic cells. As an example of an application, we consider the modeling of contaminant transport in fractured porous media. One standard model leads to a coupled system of advection–diffusion equations in a bulk (matrix) and along a surface (fracture). A series of numerical experiments with both steady and unsteady problems and different embedded geometries illustrate the numerical properties of the hybrid approach. The method demonstrates great flexibility in handling curvilinear or branching lower dimensional embedded structures.

Transient thermal performances of a salt gradient solar pond under semi-arid Moroccan climate using a 2D double-diffusive convection model

A two-dimensional numerical study is conducted to investigate quantitatively and qualitatively the thermal characteristics of a small-scale salt gradient solar pond (SGSP) operating under semi-arid climate (Marrakesh-Morocco). A realistic model was formulated based on the Navier-Stokes equations and the energy and salt transport equations. This developed model ensures an appropriate treatment of the boundary conditions, especially at the free surface of the pond in contact with the ambiance. The internal heating engendered by the absorption of solar radiation is considered in this study. The hybrid Finite-Volume Lattice-Boltzmann method is used; its performances make possible the approach of such a complex problem. After five days from the start of the heating process of the pond, the operating temperature in the heat storage zone has increased from 25°C to reach 49.4°C and the energy stored in this zone has slightly exceeded 40MJ. The storage daily efficiency of the SGSP is seen to vary in the range from 25.6% to 36.5%. The preliminary results obtained highlight the important storage potential of the experimented SGSP.

CaMEL and ADCIRC Storm Surge Models—A Comparative Study

The Computation and Modeling Engineering Laboratory (CaMEL), an implicit solver-based storm surge model, has been extended for use on high performance computing platforms. An MPI (Message Passing Interface) based parallel version of CaMEL has been developed from the previously existing serial version. CaMEL uses hybrid finite element and finite volume techniques to solve shallow water conservation equations in either a Cartesian or a spherical coordinate system and includes hurricane-induced wind stress and pressure, bottom friction, the Coriolis effect, and tidal forcing. Both semi-implicit and fully-implicit time stepping formulations are available. Once the parallel implementation is properly validated, CaMEL is evaluated against ADCIRC, an established storm surge model, using a hindcast of storm surge due to Hurricane Katrina. Observed high water marks are used to verify that both models have comparable accuracy. The effects of time step on the stability and accuracy of the models are investigated and indicate that the semi- and fully-implicit solvers in CaMEL allow the use of larger timesteps than ADCIRC’s explicit and semi-implicit solvers. However, ADCIRC outperforms CaMEL in parallel scalability and execution wall clock times. Wall times of CaMEL improve significantly when the largest stable time step sizes are used in respective models, although ADCIRC still is faster.

A hybrid finite-volume and finite difference scheme for depth-integrated non-hydrostatic model

A depth-integrated, non-hydrostatic model with hybrid finite difference and finite volume numerical algorithm is proposed in this paper. By utilizing a fraction step method, the governing equations are decomposed into hydrostatic and non-hydrostatic parts. The first part is solved by using the finite volume conservative discretization method, whilst the latter is considered by solving discretized Poisson-type equations with the finite difference method. The second-order accuracy, both in time and space, of the finite volume scheme is achieved by using an explicit predictor-correction step and linear construction of variable state in cells. The fluxes across the cell faces are computed in a Godunov-based manner by using MUSTA scheme. Slope and flux limiting technique is used to equip the algorithm with total variation dimensioning property for shock capturing purpose. Wave breaking is treated as a shock by switching off the non-hydrostatic pressure in the steep wave front locally. The model deals with moving wet/dry front in a simple way. Numerical experiments are conducted to verify the proposed model.

A second-order hybrid finite volume method for solving the Stokes equation

This paper presents a second-order hybrid finite volume method for solving the Stokes equation on a two dimensional domain. The trial function space of the method for velocity is chosen to be a quadratic conforming finite element space with a hierarchical decomposition technique on triangular meshes, and its corresponding test function space consists of piecewise constant functions and piecewise quadratic polynomial functions based on a dual partition of the domain. The trial function space and test function space of the method for pressure are chosen to be a linear finite element space. We derive the inf-sup conditions of the discrete systems of the method on triangular meshes by using a relationship between the finite volume method and the finite element method. The well-posedness of the proposed finite volume method is obtained by using the Babuska–Lax–Milgram theorem. The error estimates of the optimal order are obtained in the H1-norm for velocity and in the L2-norm for pressure. Numerical experiments are presented to illustrate the theoretical results.

Celeris: A GPU-accelerated open source software with a Boussinesq-type wave solver for real-time interactive simulation and visualization

In this paper, we introduce an interactive coastal wave simulation and visualization software, called Celeris. Celeris is an open source software which needs minimum preparation to run on a Windows machine. The software solves the extended Boussinesq equations using a hybrid finite volume–finite difference method and supports moving shoreline boundaries. The simulation and visualization are performed on the GPU using Direct3D libraries, which enables the software to run faster than real-time. Celeris provides a first-of-its-kind interactive modeling platform for coastal wave applications and it supports simultaneous visualization with both photorealistic and colormapped rendering capabilities. We validate our software through comparison with three standard benchmarks for non-breaking and breaking waves. Program summaryProgram title: CelerisProgram Files doi:http://dx.doi.org/10.17632/5djwvf5x5k.1Licensing provisions : GNU General Public License 3 (GPL)Programming language : C++, HLSLNature of problem : Boussinesq-type models provide the research-level accuracy needed for modeling wave propagation in coastal zones. However the current models, both commercial and open source, do not provide means for real-time computation, nor provide model interactivity and concurrent visualization. In order to achieve a real-time simulation speed in current parallelized models, dozens to hundreds of CPU cores are needed. Celeris is an interactive software which provides faster than real-time simulation and visualization speed on an average user laptop. The novelty of this software is its interactive environment, which allows the user to modify the model and field parameters as the model is running, and to see the effect of these changes immediately.Solution method : A hybrid finite volume-finite difference scheme is used to solve the extended Boussinesq equations. The solver is parallelized using shader programming with Direct3D libraries. Visualization is also performed with the same libraries.

Lagrangian transported MDF methods for compressible high speed flows

This paper deals with the application of thermochemical Lagrangian MDF (mass density function) methods for compressible sub- and supersonic RANS (Reynolds Averaged Navier–Stokes) simulations. A new approach to treat molecular transport is presented. This technique on the one hand ensures numerical stability of the particle solver in laminar regions of the flow field (e.g. in the viscous sublayer) and on the other hand takes differential diffusion into account. It is shown in a detailed analysis, that the new method correctly predicts first and second-order moments on the basis of conventional modeling approaches. Moreover, a number of challenges for MDF particle methods in high speed flows is discussed, e.g. high cell aspect ratio grids close to solid walls, wall heat transfer, shock resolution, and problems from statistical noise which may cause artificial shock systems in supersonic flows. A Mach 2 supersonic mixing channel with multiple shock reflection and a model rocket combustor simulation demonstrate the eligibility of this technique to practical applications. Both test cases are simulated successfully for the first time with a hybrid finite-volume (FV)/Lagrangian particle solver (PS).

ArcFVDSL, a DSEL Combined to HARTS, a Runtime System Layer to Implement Efficient Numerical Methods to Solve Diffusive Problems on New Heterogeneous Hardware Architecture

Nowadays, some frameworks like Arcane and Dune offer a number of advanced tools to deal with the complexity related to parallelism, meshes and linear solvers. However, they do not handle the high level complexity related to discretization methods and physical models. Generative programming and Domain Specific Languages (DSL) are key technologies allowing to write code with a high level expressive language and take advantage of the efficiency of generated code with low level services. DSL may be embedded in host languages like Python or C++ . Such languages, named in that case Domain Specific Embedded Languages (DSEL), are applied for instance in frameworks like Fenics or Feel++ which are dedicated to the domain of Finite Element (FE) methods and Galerkin methods. ArcFVDSL is a DSEL developed on top of the Arcane framework, aiming to implement various lowest order methods (Finite-Volume (FV), Mimetic Finite Difference (MFD), Mixed Hybrid Finite Volume (MHFV), etc.) for diffusive problems on general meshes. In this paper, we present various implementations of different complex academic problems. We focus on the capability of the language to allow the description and the resolution of these problems with several lowest-order methods. We illustrate the benefits of such technology combined to runtime system tools like Heterogeneous Abstract RunTime System (HARTS) and its ability to handle seamlessly new heterogeneous architectures with multi-core processors enhanced by General Purpose computing on Graphics Processing Units (GP-GPU). We present the performance results of each implementation on different kinds of heterogeneous hardware architecture.

An adaptive numerical method for free surface flows passing rigidly mounted obstacles

Abstract The paper develops a method for the numerical simulation of a free-surface flow of incompressible viscous fluid around a streamlined body. The body is a rigid stationary construction partially submerged in the fluid. The application we are interested in the paper is a flow around a surface mounted offshore oil platform. The numerical method builds on a hybrid finite volume / finite difference discretization using adaptive octree cubic meshes. The mesh is dynamically refined towards the free surface and the construction. Special care is taken to devise a discretization for the case of curvilinear boundaries and interfaces immersed in the octree Cartesian background computational mesh. To demonstrate the accuracy of the method, we show the results for two benchmark problems: the sloshing 3D container and the channel laminar flow passing the 3D cylinder of circular cross-section. Further, we simulate numerically a flow with surface waves around an offshore oil platform for the realistic set of geophysical data.

A Hybrid Finite Volume Method for Advection Equations and Its Applications in Population Dynamics

We present in this article a very adapted finite volume numerical scheme for transport type‐equation. The scheme is an hybrid one combining an anti‐dissipative method with down‐winding approach for the flux (Després and Lagoutière, C R Acad Sci Paris Sér I Math 328(10) (1999), 939–944; Goudon, Lagoutière, and Tine, Math Method Appl Sci 23(7) (2013), 1177–1215) and an high accurate method as the WENO5 one (Jiang and Shu, J Comput Phys 126 (1996), 202–228). The main goal is to construct a scheme able to capture in exact way the numerical solution of transport type‐equation without artifact like numerical diffusion or without “stairs” like oscillations and this for any regular or discontinuous initial distribution. This kind of numerical hybrid scheme is very suitable when properties on the long term asymptotic behavior of the solution are of central importance in the modeling what is often the case in context of population dynamics where the final distribution of the considered population and its mass preservation relation are required for prediction. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1114–1142, 2017

A hybrid Hermite-WENO/slope limiter for reconstructed discontinuous Galerkin methods on unstructured grids

Abstract In this paper, we propose a new limiter for reconstructed discontinuous Galerkin or hybrid discontinuous Galerkin and finite volume (Hybrid DG/FV) methods for solving Euler equations on two-dimensional unstructured triangular grids. The concept of hierarchical limiter which consists of two steps is introduced. The limiter firstly reconstructs a Hermite WENO polynomial by taking the advantage of the underlying DG method where the exact low-order (first-order) derivatives are available. The linear weights for the bias polynomials are optimized by a Lagrangian interpolation. To maintain the accuracy of the original methods in the smoothing region, a correction is introduced by using the second-order derivatives which are already computed in the original hybrid DG/FV methods. Then a slope limiter is used for the limitation of all the second-order derivatives. The present limiter is compact as only the von Neumann neighborhoods are required. Numerical results for both smoothing and shock contained problems are provided to validate the good performance of the present limiter.

Gradient discretization of hybrid-dimensional Darcy flow in fractured porous media with discontinuous pressures at matrix–fracture interfaces

We investigate the discretization of Darcy flow through fractured porous media on general meshes. We consider a hybrid dimensional model, invoking a complex network of planar fractures. The model accounts for matrix-fracture interactions and fractures acting either as drains or as barriers, i.e. we have to deal with pressure discontinuities at matrix-fracture interfaces. The numerical analysis is performed in the general framework of gradient discretizations which is extended to the model under consideration. Two families of schemes namely the Vertex Approximate Gradient scheme (VAG) and the Hybrid Finite Volume scheme (HFV) are detailed and shown to satisfy the gradient scheme framework, which yields, in particular, convergence. Numerical tests confirm the theoretical results. Gradient Discretization; Darcy Flow, Discrete Fracture Networks, Finite Volume

Numerical Study of Contaminant Transport in Fractured Porous Rock with Distance Dependent Dispersivity

This paper presents solute concentration profiles with constant, linear and exponential distance-dependent dispersion models for solute transport through fractured porous rock matrices. A numerical analysis has been developed for solution of the equations governing the dispersion model using the hybrid finite volume method. The results of the numerical analysis have been used to investigate the effect of the matrix diffusion coefficient, fracture velocity, and matrix and fracture retardation factors on concentration profiles. It is found that in the presence of higher matrix diffusion coefficients, the behaviour of concentration profiles is different for constant and distance-dependent dispersion models. However, the behaviour of concentration profiles is observed to be nonlinear during the small transport time period, and the matrix diffusion is found to control the spreading of solute in the fracture in the presence of constant and distance-dependent dispersion models.

Model reduction and discretization using hybrid finite volumes for flow in porous media containing faults

In this paper, we study two different model reduction strategies for solving problems involving single phase flow in a porous medium containing faults or fractures whose location and properties are known. These faults are represented as interfaces of dimension N − 1 immersed in an N dimensional domain. Both approaches can handle various configurations of position and permeability of the faults, and one can handle different fracture permeabilities on the two inner sides of the fracture. For the numerical discretization, we use the hybrid finite volume scheme as it is known to be well suited to simulating subsurface flow. Some results, which may be of use in the implementation of the proposed methods in industrial codes, are demonstrated.

Uniform temporal convergence of numerical schemes for miscible flow through porous media

The Hybrid Mimetic Mixed (HMM) family of schemes contains the Hybrid Finite Volume, Mimetic Finite Difference and Mixed Finite Volume methods. This note proves that HMM schemes, when applied to a model of the miscible displacement of one incompressible fluid by another through a porous medium, produce approximations of the concentration variable that converge uniformly towards the exact concentration.

Assessment of a hybrid finite element and finite volume code for turbulent incompressible flows

Hydra-TH is a hybrid finite-element/finite-volume incompressible/low-Mach flow simulation code based on the Hydra multiphysics toolkit being developed and used for thermal-hydraulics applications. In the present work, a suite of verification and validation (V&V) test problems for Hydra-TH was defined to meet the design requirements of the Consortium for Advanced Simulation of Light Water Reactors (CASL). The intent for this test problem suite is to provide baseline comparison data that demonstrates the performance of the Hydra-TH solution methods. The simulation problems vary in complexity from laminar to turbulent flows. A set of RANS and LES turbulence models were used in the simulation of four classical test problems. Numerical results obtained by Hydra-TH agreed well with either the available analytical solution or experimental data, indicating the verified and validated implementation of these turbulence models in Hydra-TH. Where possible, some form of solution verification has been attempted to identify sensitivities in the solution methods, and suggest best practices when using the Hydra-TH code. • We performed a comprehensive study to verify and validate the turbulence models in Hydra-TH. • Hydra-TH delivers 2nd-order grid convergence for the incompressible Navier–Stokes equations. • Hydra-TH can accurately simulate the laminar boundary layers. • Hydra-TH can accurately simulate the turbulent boundary layers with RANS turbulence models. • Hydra-TH delivers high-fidelity LES capability for simulating turbulent flows in confined space.

Gradient discretization of hybrid dimensional Darcy flows in fractured porous media

This article deals with the discretization of hybrid dimensional Darcy flows in fractured porous media. These models couple the flow in the fractures represented as surfaces of codimension one with the flow in the surrounding matrix. The convergence analysis is carried out in the framework of gradient schemes which accounts for a large family of conforming and nonconforming discretizations. The vertex approximate gradient scheme and the hybrid finite volume scheme are extended to such models and are shown to verify the gradient scheme framework. Our theoretical results are confirmed by numerical experiments performed on tetrahedral, Cartesian and hexahedral meshes in heterogeneous isotropic and anisotropic porous media.

A two-dimensional depth-integrated non-hydrostatic numerical model for nearshore wave propagation

In this study, we develop a shallow-water depth-integrated non-hydrostatic numerical model (SNH model) using a hybrid finite-volume and finite-difference method. Numerical discretization is performed using the non-incremental pressure-correction method on a collocated grid. We demonstrate that an extension can easily be made from an existing finite-volume method and collocated-grid based hydrostatic shallow-water equations (SWE) model to a non-hydrostatic model. A series of benchmark tests are used to validate the proposed numerical model. Our results demonstrate that the proposed model is robust and well-balanced, and it captures the wet–dry fronts accurately. A comparison between the SNH and SWE models indicates the importance of considering the wave dispersion effect in simulations when the wave amplitude to water depth ratio is large.

A new numerical model for simulations of wave transformation, breaking and long‐shore currents in complex coastal regions

SummaryIn this paper, we propose a model based on a new contravariant integral form of the fully nonlinear Boussinesq equations in order to simulate wave transformation phenomena, wave breaking, and nearshore currents in computational domains representing the complex morphology of real coastal regions. The aforementioned contravariant integral form, in which Christoffel symbols are absent, is characterized by the fact that the continuity equation does not include any dispersive term. A procedure developed in order to correct errors related to the difficulties of numerically satisfying the metric identities in the numerical integration of fully nonlinear Boussinesq equation on generalized boundary‐conforming grids is presented. The Boussinesq equation system is numerically solved by a hybrid finite volume–finite difference scheme. The proposed high‐order upwind weighted essentially non‐oscillatory finite volume scheme involves an exact Riemann solver and is based on a genuinely two‐dimensional reconstruction procedure, which uses a convex combination of biquadratic polynomials. The wave breaking is represented by discontinuities of the weak solution of the integral form of the nonlinear shallow water equations.The capacity of the proposed model to correctly represent wave propagation, wave breaking, and wave‐induced currents is verified against test cases present in the literature. The results obtained are compared with experimental measures, analytical solutions, or alternative numerical solutions. Copyright © 2015 John Wiley & Sons, Ltd.