Abstract. In this paper, we apply a semi-implicit **finite** **volume** **method** for the numerical simulation of density driven flows in porous media; this amounts to solving a nonlinear convection-diffusion par- abolic equation for the concentration coupled with an elliptic equation for the pressure. We compute the solutions for two specific problems: a problem involving a rotating interface between salt and fresh water and the classical but difficult Henry’s problem. All solutions are compared to results obtained by running FEflow, a commercial software package for the simulation of groundwater flow, mass and heat transfer in porous media.

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In this paper, a review of **finite** **volume** methods has been made. Based on the traditional numerical fluxes, the Local Lax-friedrichs and the Lax-Wendroff fluxes, a new numerical flux and consequently a new **finite** **volume** **method** were constructed. It was proved that the proposed new numerical **method** satisfies the properties of a **finite** **volume** **method**. Several challenging test problems, especially those with discontinuities solutions for Euler Equations,

The Cell Centred **Finite** **Volume** **method** represent the numerical scheme which is able to solve the flow problem pass through nozzle. Different nozzle geometry gives different flow behaviour. Particular shape of nozzle may generate the flow inside nozzle goes to subsonic solution if the pressure ratio between entry and exit station is not having a sufficient value. Considering the result as shown in Fig.2, Fig.3 and Fig.5, it is clear indicated that the Cell Centred **Finite** **Volume** able to capture the presence of shock wave inside the nozzle without smearing or oscillatory solution near the shock. Therefore this approach can be considered as an appropriate approach for solving fluid flow problem from quasi one dimensional flow problems to two dimensional flow problem such as flow pass through airfoil or other two dimensional flow problems.

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In this paper, evaluating a damaged member in Timoshenko beam, applying the **finite** **volume** **method** (FVM) has been examined. Damage identification of beams using a mode shape (or displacement) based indicator (MSBI or DBI) has been studied. The efficiency of the FVM based damage indicator has been assessed with considering a simply supported beam having different characteristics for static, buckling, and free vibration analysis. As it’s presented in the numerical instances , comparing the acquired results by **finite** **volume** **method** with the same procedure extracted from **finite** element **method** indicates a good match between the two methods, and there are rational correlations between them . Consequently, the **finite** **volume** **method** can be precisely applied for damage localization in the beam like structures. It was also presented that FVM can show the damaged element for both thin and thick beams without observing shear locking while shear locking is observed in the analysis of thin beams using FEM.

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Abstract—A hybrid time-domain **method** combing **finite**-difference and cell-centered **finite**-**volume** **method** is presented in this paper. This **method** is applied to solve three dimensional electromagnetic problems which involve media having **finite** conductivity. The fractional-step technique (FST) for FVTD scheme is applied to solve these problems. Local time-step scheme is used to enhance the efficiency of this **method**. Numerical results are given and compared with a reliable numerical **method**, which is used to show the validation of this **method**.

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There are several techniques used to parallelize the iterative linear solver such as GPU-CUDA, MPI, OpenMP, hybrid methods, etc. As an initial parallel approach, in this paper, we propose a Jacobi **method** in which all the calculations were performed using several kernels in a GPU-CUDA platform. For each kernel, we are allowed to configure several aspects of the grid, including the number of threads per block, as well as the number of blocks. In the proposed algorithm, control volumes are processed in rectangular blocks of data assigned to threads whose configuration was chosen to take advantage of the element distribution of the problem and easy implementation of the code. Therefore, the novelty of this work is to present a strategy and the results of our effort to exploit the computational capabilities of GPUs under the CUDA environment in order to solve the 3D Poisson equation using a **finite** **volume** **method** on arbitrary geometries.

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On the one hand, commercial software is often based on the **Finite**-Element-**Method** (FEM) to simulate the RRIM process [8–11]. On the other hand, **Finite**-**Volume**-based solvers, representing the state of research, focus on thermoplastic injection molding using incompressible and isothermal models [12]. In this study, the **Finite**-**Volume**-**Method** (FVM) is used, resolving the flux at the cell faces and using an Eulerian approach [13–15]. Due to this flow modeling, FVM provides a more realistic multiphase flow with physical significance on the fluxes, which is not possible in FEM by solving a Lagrangian mesh at the nodes. For implementation and simulation, the open source Computational-Fluid-Dynamics (CFD) toolbox OpenFOAM 4.1 (OpenCFD Ltd., Bracknell, UK) [7,12–15] is used and well-known viscosity, curing and fiber orientation models are implemented to model the reinforced reactive injection molding process. A solver for compressible, non-isothermal multiphase flow is extended, using a phase depending boundary condition, defined to enable mold-filling simulation, by separating and interpolating boundary conditions for polymer and air. Additionally, the solver predicts fiber orientation, resulting from the flow during mold filling.

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During irradiation of the tissue by laser, different types of interaction may occur, among which thermal effects have a larger contribution. Therefore, modeling this effect enables us to predict the optimal laser wavelength, pulse width, and irradiance power. Also, it helps us to predict the outcome of the surgery, which is very desirable as it facilitates the process of treatment. In fact laser parameters and tissue characteristics dominate the process of laser tissue interaction. Optical behavior of tissue has a strong dependency on laser wavelength, while heat transportation is merely directed by tissue thermal characteristics [1]. Heat transport process is defined by a differential equation which needs to be solved numerically. In order to solve these equations, which describe dynamic systems, different techniques have been adopted. **Finite** **volume**, which was defined based on **finite** difference **method**, was first introduced by Borris and Brook in 1973 [1]. In this paper implicit **finite** **volume** **method** is used to solve heat transport equation, as it shows higher stability criteria than those used formerly [2, 3, 4].

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In this extended form of FVM, the MLS technique is used for the construction of interpolation functions for the integration points considered on cell boundaries. Several problems have been solved by this **method** and the results have been compared with the analytical, numerical and experimental results. The comparisons have shown that the predictions of the present **method** have good agreement with the reference results for SIF calculations and crack growth path predictions. One of the interesting features of the presented **finite** **volume** **method** is that in each step of analysis, due to crack growth, only cell dividing near the crack tip is needed which can be implemented easily rather than extensive remeshing which is needed in the **finite** element **method**. Also, the present study reveals the great potential of FVM for the fracture analysis of structures with a prospect of application for the existed challenges in the dynamic crack propagation.

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Abstract. This paper investigates the supra-convergence phenomenon that one can observe in the upwind **finite** **volume** **method** for solving linear convection problem on a bounded domain but also in **finite** difference scheme with non-uniform grids. Although the scheme is no longer consistent in the **finite** difference sense and Lax-Richtmyer theorem not suitable, it is a well-known convergent **method**. In order to analyze the convergence rate, we introduce what we call a geometric corrector, which is associated with every **finite** **volume** mesh and every constant convection vector. Under a local quasi- uniformity condition and if the continuous solution is regular enough, there is a link between the convergence of the **finite** **volume** scheme and this geometric corrector : the study of this latter leads to the proof of the optimal order of convergence. We then focus our attention on an uniformly refined mesh of quadrangles and on a series of independent meshes of triangles and tetrahedrons. In these latter cases, a loss of accuracy is observed if there exists in the family of meshes a fixed straight line parallel to the convection direction.

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We have analyzed the suitability of the **finite**-**volume** **method** for calculating incompressible, viscous, non-Newtonian fluid flow where the Sisko model was used. In addition, convergence criteria are presented and the convergence depending on the mesh size was analyzed. The **method** was tested for the driven-cavity case and flow in a channel with a sudden contraction. The numerical solution was compared with the results available in the open literature. For the rheological model, parameters obtained from an experiment with a capillary viscometer were used.

In recent years, a great deal of effort has been made in developing **finite** **volume** methods to solve the compressible Navier-Stokes equations on unstructured grids. There exist two major classes of the **finite** **volume** methods: cell-centered and vertex-centered **finite** **volume** methods. A vertex-centered **finite** **volume** **method** is more popular when used for discretizing the viscous fluxes in the Navier-Stokes equations, since it avoids the difficulty for the discretization of the viscous fluxes by using a hybrid formulation, where the discretization of the inviscid fluxes is carried out via vertex-centered **finite** **volume** and **finite** element approximation is used for the discretization of the viscous fluxes. The cell-centered **finite** **volume** **method** is also widely used to solve compressible Navier-Stokes equations in engineering due to the satisfaction of the integral form of the conservation law and its simplicity of implementation. However, it does not possess the natural tools for gradient calculation and interpolation [7]. In the case of structured grids, the discretization of the diffusive fluxes is straightforward, where the gradients are usually estimated by central differencing in the corresponding curvilinear coordinates direction. However, the approximation of gradients using a cell-centered **finite** **volume** **method** is considerably more complicated on unstructured grids.

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This paper developed a new numerical model for simulating overland flow using fully dynamic shallow water equations based on the unstructured **finite** **volume** **method**. The sheet flow regime is used when calculating the numerical flux and discretizing the bottom slope source terms. In order to reduce the numerical instability induced by very shallow water depth often occurred in overland flows, the semi-implicit scheme is used to deal with the slope friction source terms. This two- dimensional numerical model allows a direct transformation of the rainfall into the runoff hydrograph at the catchment outlet. The model has been verified by reproducing the analytical and experimental results in one-dimensional and two-dimensional situations and the results prove that it has more comprehensive calculation capabilities and good robust performances. In the future research, the model will be used to simulate the rainfall-runoff process of the real natural catchment to validate its practicability.

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In this section we assess the capabilities of the **finite**-**volume** **method** described previously by presenting results of simulations for the benchmark flow through a 4:1 planar sudden contraction shown in Figure 5 under conditions of negligible inertia. This is a long standing classic benchmark in computational rheology (Hassager, 1988), where the difficulty lies at the correct prediction of the large stresses and stress gradients in the vicinity of the re- entrant corner (generally all models show the stresses to grow to infinity as the corner is approached) making this flow very sensitive to highly elastic flows. In particular, it is important to know the upstream vortex growth mechanisms due to flow elasticity, and the corresponding large pressure drops and overshoot of the axial velocity along the centreline.

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This paper describes a numerical solution for two dimensional partial differential equations modeling (or arising from) a fluid flow and transport phenomena. The diffusion equation is discretized by the Nodal methods. The saturation equation is solved by a **finite** **volume** **method**. We start with incompressible sin- gle-phase flow and move step-by-step to the black-oil model and compressible two phase flow. Numerical results are presented to see the performance of the **method**, and seem to be interesting by comparing them with other recent results.

Abstract In this paper, fitted **finite** **volume** **method** is developed to solve a nonlinear degenerate Black-Scholes equation applied in the valuation of unit-linked policy with surrender option, based on the fitting idea in S. Wang [IMA J. Numer. Anal., 24 (2004), 699–720]. Unlike the conventional pricing **method** mentioned in [1] which is using the free boundary **method** to calibrate the valuation PDE, here we develop a power penalty **method** to solve numerically the linear complimentary problem in the variational inequality arising from the valuation of unit-linked policy with surren- der option. With the degenerate boundary and non-smooth final condition, we will show that it is essential to refine the mesh to remain the convergence and super-convergence order.

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Considering an Eulerian point of view, the ow eld unknowns can then be described as the functions _m(x; t), P (x; t), (x; t) and T (x; t). The spatial domain of the pulse tube is divided to n nite volumes. Thus there are n + 1 faces as shown in Figure 2. Solid circles (1 I n) store the nodal values of the temperature, pressure, density, mass, and gas properties. The cross signs (1 i n + 1) are the storage locations for the nodal values of the mass ow rate, velocity, enthalpy ow, energy ow, and the exergy ow. This staggered arrangement of nite volumes helps to prevent non- physical numerical solutions in the **Finite** **Volume** **Method** (FVM).

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This research deals with the development of a Computational Fluid Dynamics (CFD) code called V-Flow using MATLAB for the two-dimensional modelling of unsteady flow over stepped spillways. V-Flow can be coupled with GAMBIT software and different spillway geometries can be modelled using voronoi mesh elements. The governing equations of flow over stepped spillways were discretized using the **Finite** **Volume** **Method** (FVM). The Power-Law scheme, implicit time approximation, Gauss–Seidel **method**, and SIMPLE algorithm were used for the discretization procedure. The flow was considered to be a laminar fluid flow with no turbulence model. The V-Flow model was validated against velocity vectors, streamlines, static pressures, dynamic pressures, and total pressures over the spillway obtained from the FLUENT model application. The experimental model simulated using both V-Flow and FLUENT for validation according to Gonzalez’s experiments. A comparison between the results obtained from the application of the different models showed good agreement according to the mass imbalance, which serves as both a useful indicator of the convergence of the fluid flow solution and as an accuracy criterion to compare the V-Flow and FLUENT results.

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In his Master's thesis, R. Hiemstra, 2011, [25] presented an interesting combination of IGA with a mimetic discretization **method**. The latter is a fusion of concepts of FEA and **finite** **volume** **method** (FVM). The resulting scheme, using B-splines, resembles a FVM on a staggered grid for the representation of the conservation laws and FEA for the representation of the constitutive equations. It allows a decomposition of the field variable within the global topology. This characteristic was exploited to numerically determine the lift produced by an irrotational, incompressible flow over an airfoil. It was observed that accurate results were obtained even with quite coarse meshes and that this approach leads to stable and consistent estimations of the lift, even under mesh distortion on the trailing edge.

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