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Статті в журналах з теми "Numerical analysis : finite volumes"

Автор: Grafiati

Опубліковано: 15 березня 2025

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1

Idelsohn, S. R., and E. Oñate. "Finite volumes and finite elements: Two ‘good friends’." International Journal for Numerical Methods in Engineering 37, no. 19 (October 15, 1994): 3323–41. http://dx.doi.org/10.1002/nme.1620371908.

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2

Droniou, Jérôme, Robert Eymard, Thierry Gallouët, and Raphaèle Herbin. "The Gradient Discretisation Method for Linear Advection Problems." Computational Methods in Applied Mathematics 20, no. 3 (July 1, 2020): 437–58. http://dx.doi.org/10.1515/cmam-2019-0060.

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Анотація:

AbstractWe adapt the Gradient Discretisation Method (GDM), originally designed for elliptic and parabolic partial differential equations, to the case of a linear scalar hyperbolic equations. This enables the simultaneous design and convergence analysis of various numerical schemes, corresponding to the methods known to be GDMs, such as finite elements (conforming or non-conforming, standard or mass-lumped), finite volumes on rectangular or simplicial grids, and other recent methods developed for general polytopal meshes. The scheme is of centred type, with added linear or non-linear numerical diffusion. We complement the convergence analysis with numerical tests based on the mass-lumped {\mathbb{P}_{1}} conforming and non-conforming finite element and on the hybrid finite volume method.

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3

Khattri, Sanjay Kumar. "Nonlinear elliptic problems with the method of finite volumes." Differential Equations and Nonlinear Mechanics 2006 (2006): 1–16. http://dx.doi.org/10.1155/denm/2006/31797.

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We present a finite volume discretization of the nonlinear elliptic problems. The discretization results in a nonlinear algebraic system of equations. A Newton-Krylov algorithm is also presented for solving the system of nonlinear algebraic equations. Numerically solving nonlinear partial differential equations consists of discretizing the nonlinear partial differential equation and then solving the formed nonlinear system of equations. We demonstrate the convergence of the discretization scheme and also the convergence of the Newton solver through a variety of practical numerical examples.

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4

Dubois, Fran�ois. "Finite volumes and mixed Petrov-Galerkin finite elements: The unidimensional problem." Numerical Methods for Partial Differential Equations 16, no. 3 (May 2000): 335–60. http://dx.doi.org/10.1002/(sici)1098-2426(200005)16:3<335::aid-num5>3.0.co;2-x.

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5

Da Silva Almeida Junior, Dilberto, Anderson de Jesus Araujo Ramos, Joao Carlos Pantoja Fortes, and Mauro De Lima Santos. "Ingham type approach for uniform observability inequality of the semi-discrete coupled wave equations." Electronic Journal of Differential Equations 2020, no. 01-132 (December 22, 2020): 127. http://dx.doi.org/10.58997/ejde.2020.127.

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Анотація:

This article concerns an observability inequality for a system of coupled wave equations for the continuous models as well as for the space semi-discrete finite difference approximations. For finite difference and standard finite elements methods on uniform numerical meshes it is known that a numerical pathology produces a blow-up of the constant on the observability inequality as the mesh-size tends to zero. We identify this numerical anomaly for coupled wave equations and we prove that there exists a uniform observability inequality in a subspace of solutions generated by low frequencies. We use the Ingham type approach for getting a uniform boundary observability. For more information see https://ejde.math.txstate.edu/Volumes/2020/127/abstr.html

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6

LIU, S. J., H. WANG, and H. ZHANG. "SMOOTHED FINITE ELEMENTS LARGE DEFORMATION ANALYSIS." International Journal of Computational Methods 07, no. 03 (September 2010): 513–24. http://dx.doi.org/10.1142/s0219876210002246.

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Анотація:

The smoothed finite element method (SFEM) was developed in order to eliminate certain shortcomings of the finite element method (FEM). SFEM enjoys some of the flexibilities of meshfree methods. One advantage of SFEM is its applicability to modeling large deformations. Due to the absence of volume integration and parametric mapping, issues such as negative volumes and singular Jacobi matrix do not occur. However, despite these advantages, SFEM has never been applied to problems with extreme large deformation. For the first time, we apply SFEM to extreme large deformations. For two numerical problems, we demonstrate the advantages of SFEM over FEM. We also show that SFEM can compete with the flexibility of meshfree methods.

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7

Diniz, Jacqueline F. B., João M. P. Q. Delgado, Anderson F. Vilela, Ricardo S. Gomez, Arianne D. Viana, Maria J. Figueiredo, Diego D. S. Diniz, et al. "Drying of Sisal Fiber: A Numerical Analysis by Finite-Volumes." Energies 14, no. 9 (April 27, 2021): 2514. http://dx.doi.org/10.3390/en14092514.

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Анотація:

Vegetable fibers have inspired studies in academia and industry, because of their good characteristics appropriated for many technological applications. Sisal fibers (Agave sisalana variety), when extracted from the leaf, are wet and must be dried to reduce moisture content, minimizing deterioration and degradation for long time. The control of the drying process plays an important role to guarantee maximum quality of the fibers related to mechanical strength and color. In this sense, this research aims to evaluate the drying of sisal fibers in an oven with mechanical air circulation. For this purpose, a transient and 3D mathematical model has been developed to predict moisture removal and heating of a fiber porous bed, and drying experiments were carried out at different drying conditions. The advanced model considers bed porosity, fiber and bed moisture, simultaneous heat and mass transfer, and heat transport due to conduction, convection and evaporation. Simulated drying and heating curves and the hygroscopic equilibrium moisture content of the sisal fibers are presented and compared with the experimental data, and good concordance was obtained. Results of moisture content and temperature distribution within the fiber porous bed are presented and discussed in details. It was observed that the moisture removal and temperature kinetics of the sisal fibers were affected by the temperature and relative humidity of the drying air, being more accentuated at higher temperature and lower relative humidity, and the drying process occurred in a falling rate period.

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8

Deuring, Paul, and Robert Eymard. "L2-stability of a finite element – finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions." ESAIM: Mathematical Modelling and Numerical Analysis 51, no. 3 (April 14, 2017): 919–47. http://dx.doi.org/10.1051/m2an/2016042.

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Анотація:

We consider a time-dependent and a steady linear convection-diffusion-reaction equation whose coefficients are nonconstant. Boundary conditions are mixed (Dirichlet and Robin−Neumann) and nonhomogeneous. Both the unsteady and the steady problem are approximately solved by a combined finite element – finite volume method: the diffusion term is discretized by Crouzeix−Raviart piecewise linear finite elements on a triangular grid, and the convection term by upwind barycentric finite volumes. In the unsteady case, the implicit Euler method is used as time discretization. This scheme is shown to be unconditionally L2-stable, uniformly with respect to diffusion, except if the Robin−Neumann boundary condition is inhomogeneous and the convective velocity is tangential at some points of the Robin−Neumann boundary. In that case, a negative power of the diffusion coefficient arises. As is shown by a counterexample, this exception cannot be avoided.

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9

Droniou, Jérome, Neela Nataraj, and Devika Shylaja. "Numerical Analysis for the Pure Neumann Control Problem Using the Gradient Discretisation Method." Computational Methods in Applied Mathematics 18, no. 4 (October 1, 2018): 609–37. http://dx.doi.org/10.1515/cmam-2017-0054.

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Анотація:

AbstractThe article discusses the gradient discretisation method (GDM) for distributed optimal control problems governed by diffusion equation with pure Neumann boundary condition. Using the GDM framework enables to develop an analysis that directly applies to a wide range of numerical schemes, from conforming and non-conforming finite elements, to mixed finite elements, to finite volumes and mimetic finite differences methods. Optimal order error estimates for state, adjoint and control variables for low-order schemes are derived under standard regularity assumptions. A novel projection relation between the optimal control and the adjoint variable allows the proof of a super-convergence result for post-processed control. Numerical experiments performed using a modified active set strategy algorithm for conforming, non-conforming and mimetic finite difference methods confirm the theoretical rates of convergence.

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10

Graff, Joseph S., Roger L. Davis, and John P. Clark. "Computational structural dynamics general solution procedure using finite volumes." Journal of Algorithms & Computational Technology 16 (January 2022): 174830262210840. http://dx.doi.org/10.1177/17483026221084030.

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Анотація:

A method for the solution of the three-dimensional structural dynamics equations with large strains using a finite volume technique is presented. The proposed solution procedure is second order accurate in space and employs a second-order accurate dual time-stepping scheme. The momentum conservation equations are written in terms of the Piola-Kirchhoff stresses. The stress tensor is related to the Lagrangian strain tensor through the St. Venant-Kirchhoff constitutive relationship. The structural solver presented is verified through two test cases. The first test case is a three-dimensional cantilever beam subject to a gravitational load that is verified using theory and two-dimensional simulations reported in literature. The second test case is a three-dimensional highly deformable cantilever plate subject to a gravitational load. The results of this case are verified through a comparison with the modal response calculated by commercially available software. The focus of the current effort is the development and verification of the structural dynamics portion of a future fully coupled monolithic fluid-thermal-structure interaction code package.

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11

Casadei, Folco, and Nicolas Leconte. "Coupling finite elements and finite volumes by Lagrange multipliers for explicit dynamic fluid-structure interaction." International Journal for Numerical Methods in Engineering 86, no. 1 (October 28, 2010): 1–17. http://dx.doi.org/10.1002/nme.3042.

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12

Chhetri, Maya, Petr Girg, and Elliott Hollifield. "Existence of positive solutions for fractional Laplacian equations: theory and numerical experiments." Electronic Journal of Differential Equations 2020, no. 01-132 (July 28, 2020): 81. http://dx.doi.org/10.58997/ejde.2020.81.

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Анотація:

We consider a class of nonlinear fractional Laplacian problems satisfying the homogeneous Dirichlet condition on the exterior of a bounded domain. We prove the existence of positive weak solution for classes of sublinear nonlinearities including logistic type. A method of sub- and supersolution, without monotone iteration, is established to prove our existence results. We also provide numerical bifurcation diagrams and the profile of positive solutions, corresponding to the theoretical results using the finite element method in one dimension. For more information see https://ejde.math.txstate.edu/Volumes/2020/81/abstr.html

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13

Margolin, L. G. "Finite-scale equations for compressible fluid flow." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 367, no. 1899 (July 28, 2009): 2861–71. http://dx.doi.org/10.1098/rsta.2008.0290.

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Анотація:

Finite-scale equations (FSE) describe the evolution of finite volumes of fluid over time. We discuss the FSE for a one-dimensional compressible fluid, whose every point is governed by the Navier–Stokes equations. The FSE contain new momentum and internal energy transport terms. These are similar to terms added in numerical simulation for high-speed flows (e.g. artificial viscosity) and for turbulent flows (e.g. subgrid scale models). These similarities suggest that the FSE may provide new insight as a basis for computational fluid dynamics. Our analysis of the FS continuity equation leads to a physical interpretation of the new transport terms, and indicates the need to carefully distinguish between volume-averaged and mass-averaged velocities in numerical simulation. We make preliminary connections to the other recent work reformulating Navier–Stokes equations.

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14

Lazarov, R. D., Ilya D. Mishev, and P. S. Vassilevski. "Finite Volume Methods for Convection-Diffusion Problems." SIAM Journal on Numerical Analysis 33, no. 1 (February 1996): 31–55. http://dx.doi.org/10.1137/0733003.

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15

Maury, Bertrand. "Numerical Analysis of a Finite Element/Volume Penalty Method." SIAM Journal on Numerical Analysis 47, no. 2 (January 2009): 1126–48. http://dx.doi.org/10.1137/080712799.

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16

Feistauer, Miloslav, Jirí Felcman, Mária Lukácová-Medvid'ová, and Gerald Warnecke. "Error Estimates for a Combined Finite Volume--Finite Element Method for Nonlinear Convection--Diffusion Problems." SIAM Journal on Numerical Analysis 36, no. 5 (January 1999): 1528–48. http://dx.doi.org/10.1137/s0036142997314695.

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17

Clain, Stéphane. "Finite Volume Maximum Principle for Hyperbolic Scalar Problems." SIAM Journal on Numerical Analysis 51, no. 1 (January 2013): 467–90. http://dx.doi.org/10.1137/110854278.

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18

Zou, Qingsong, Li Guo, and Quanling Deng. "High Order Continuous Local-Conserving Fluxes and Finite-Volume-Like Finite Element Solutions for Elliptic Equations." SIAM Journal on Numerical Analysis 55, no. 6 (January 2017): 2666–86. http://dx.doi.org/10.1137/16m1066567.

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19

Fabbri, Giampietro, Matteo Greppi, and Federico Amati. "Numerical Analysis of New PCM Thermal Storage Systems." Energies 17, no. 7 (April 8, 2024): 1772. http://dx.doi.org/10.3390/en17071772.

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In this paper, a thermal storage system based on a phase change material is proposed and investigated. The system is composed of several tubes that cross a phase change material mass. A fluid flowing in the tubes charges and discharges the heat storage system. A mathematical model of the system has been developed, which provides the time and space distribution of velocity, temperature, and liquid phase-changing material concentration in a non-stationary regime. A hybrid solution method based on finite volumes and finite differences techniques has been employed for the model equations in the MATLAB environment. To the tubes, a rectangular cross section has been assigned. The performance of the system in terms of accumulated energy density and accumulated power density has been investigated by varying some geometric parameters. The considered geometric parameters influence the number of tubes per unit of system width, the tube hydraulic resistance, the amount of phase change material around each tube, the heat transfer surface of the tube, and the heat storage velocity. In the parametric analysis, peaks have been evidenced in the investigated performance parameters at different instants after the beginning of the heat storage.

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20

Jo, Gwanghyun, and Do Y. Kwak. "Immersed finite element methods for convection diffusion equations." AIMS Mathematics 8, no. 4 (2023): 8034–59. http://dx.doi.org/10.3934/math.2023407.

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Анотація:

<abstract><p>In this work, we develop two IFEMs for convection-diffusion equations with interfaces. We first define bilinear forms by adding judiciously defined convection-related line integrals. By establishing Gårding's inequality, we prove the optimal error estimates both in $ L^2 $ and $ H^1 $-norms. The second method is devoted to the convection-dominated case, where test functions are piecewise constant functions on vertex-associated control volumes. We accompany the so-called upwinding concepts to make the control-volume based IFEM robust to the magnitude of convection terms. The $ H^1 $ optimal error estimate is proven for control-volume based IFEM. We document numerical experiments which confirm the analysis.</p></abstract>

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21

Sonar, Thomas. "On Families of Pointwise Optimal Finite Volume ENO Approximations." SIAM Journal on Numerical Analysis 35, no. 6 (December 1998): 2350–69. http://dx.doi.org/10.1137/s0036142997316013.

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22

Glitzky, Annegret, and Jens A. Griepentrog. "Discrete Sobolev–Poincaré Inequalities for Voronoi Finite Volume Approximations." SIAM Journal on Numerical Analysis 48, no. 1 (January 2010): 372–91. http://dx.doi.org/10.1137/09076502x.

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23

Gallouet, T., and J. P. Vila. "Finite Volume Schemes for Conservation Laws of Mixed Type." SIAM Journal on Numerical Analysis 28, no. 6 (December 1991): 1548–73. http://dx.doi.org/10.1137/0728079.

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24

Morton, K. W. "On the Analysis of Finite Volume Methods for Evolutionary Problems." SIAM Journal on Numerical Analysis 35, no. 6 (December 1998): 2195–222. http://dx.doi.org/10.1137/s0036142997316967.

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25

Cui, Ming, and Xiu Ye. "Unified Analysis of Finite Volume Methods for the Stokes Equations." SIAM Journal on Numerical Analysis 48, no. 3 (January 2010): 824–39. http://dx.doi.org/10.1137/090780985.

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26

Rusanov, P. G. "Algorithmic concepts of method of solid bodies." Izvestiya MGTU MAMI 7, no. 3-1 (February 10, 2013): 124–36. http://dx.doi.org/10.17816/2074-0530-68053.

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Анотація:

Method of solid bodies is included in the group of adaptive numerical methods of continuum mechanics. It is applicable for the analysis of motion of solids, liquids and gases. Using construction techniques this method provides a priori division of the generalized coordinates for the finite object of the research volume into fast and slow variables. The total number of slow variables does not exceed 6 N, where N is a number of finite volumes. The paper mentions methods of forming mathematical model of the object state relatively to the slow variables without the participation of the fast variables.

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27

El Moutea, Omar, and Hassan El Amri. "Combined mixed finite element and nonconforming finite volume methods for flow and transport in porous media." Analysis 41, no. 3 (July 23, 2021): 123–44. http://dx.doi.org/10.1515/anly-2018-0019.

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Abstract This paper is concerned with numerical methods for a coupled system of two partial differential equations (PDEs), modeling flow and transport of a contaminant in porous media. This coupled system, arising in modeling of flow and transport in heterogeneous porous media, includes two types of equations: an elliptic and a diffusion-convection equation. We focus on miscible flow in heterogeneous porous media. We use the mixed finite element method for the Darcy flow equation over triangles, and for the concentration equation, we use nonconforming finite volume methods in unstructured mesh. Finally, we show the existence and uniqueness of a solution of this coupled scheme and demonstrate the effectiveness of the methodology by a series of numerical examples.

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28

Brocchini, Maurizio. "A reasoned overview on Boussinesq-type models: the interplay between physics, mathematics and numerics." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469, no. 2160 (December 8, 2013): 20130496. http://dx.doi.org/10.1098/rspa.2013.0496.

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This paper, which is largely the fruit of an invited talk on the topic at the latest International Conference on Coastal Engineering, describes the state of the art of modelling by means of Boussinesq-type models (BTMs). Motivations for using BTMs as well as their fundamentals are illustrated, with special attention to the interplay between the physics to be described, the chosen model equations and the numerics in use. The perspective of the analysis is that of a physicist/engineer rather than of an applied mathematician. The chronological progress of the currently available BTMs from the pioneering models of the late 1960s is given. The main applications of BTMs are illustrated, with reference to specific models and methods. The evolution in time of the numerical methods used to solve BTMs (e.g. finite differences, finite elements, finite volumes) is described, with specific focus on finite volumes. Finally, an overview of the most important BTMs currently available is presented, as well as some indications on improvements required and fields of applications that call for attention.

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29

FRANCHI, C. G., and F. MONTELAGHI. "A WEAK-WEAK FORMULATION FOR LARGE DISPLACEMENTS BEAM STATICS: A FINITE VOLUMES APPROXIMATION." International Journal for Numerical Methods in Engineering 39, no. 4 (February 28, 1996): 585–604. http://dx.doi.org/10.1002/(sici)1097-0207(19960229)39:4<585::aid-nme871>3.0.co;2-f.

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30

Du, Qiang, and Lili Ju. "Finite Volume Methods on Spheres and Spherical Centroidal Voronoi Meshes." SIAM Journal on Numerical Analysis 43, no. 4 (January 2005): 1673–92. http://dx.doi.org/10.1137/s0036142903425410.

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31

Lenz, Martin, Simplice Firmin Nemadjieu, and Martin Rumpf. "A Convergent Finite Volume Scheme for Diffusion on Evolving Surfaces." SIAM Journal on Numerical Analysis 49, no. 1 (January 2011): 15–37. http://dx.doi.org/10.1137/090776767.

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32

Halpern, Laurence, and Florence Hubert. "A Finite Volume Ventcell-Schwarz Algorithm for Advection-Diffusion Equations." SIAM Journal on Numerical Analysis 52, no. 3 (January 2014): 1269–91. http://dx.doi.org/10.1137/130919799.

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33

Cockburn, B., F. Coquel, and P. G. LeFloch. "Convergence of the Finite Volume Method for Multidimensional Conservation Laws." SIAM Journal on Numerical Analysis 32, no. 3 (June 1995): 687–705. http://dx.doi.org/10.1137/0732032.

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34

Coudière, Yves, and Gianmarco Manzini. "The Discrete Duality Finite Volume Method for Convection-diffusion Problems." SIAM Journal on Numerical Analysis 47, no. 6 (January 2010): 4163–92. http://dx.doi.org/10.1137/080731219.

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35

Hussain, Arafat, Zhoushun Zheng, and Eyaya Fekadie Anley. "Numerical Analysis of Convection–Diffusion Using a Modified Upwind Approach in the Finite Volume Method." Mathematics 8, no. 11 (October 28, 2020): 1869. http://dx.doi.org/10.3390/math8111869.

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Анотація:

The main focus of this study was to develop a numerical scheme with new expressions for interface flux approximations based on the upwind approach in the finite volume method. Our new proposed numerical scheme is unconditionally stable with second-order accuracy in both space and time. The method is based on the second-order formulation for the temporal approximation, and an upwind approach of the finite volume method is used for spatial interface approximation. Some numerical experiments have been conducted to illustrate the performance of the new numerical scheme for a convection–diffusion problem. For the phenomena of convection dominance and diffusion dominance, we developed a comparative study of this new upwind finite volume method with an existing upwind form and central difference scheme of the finite volume method. The modified numerical scheme shows highly accurate results as compared to both numerical schemes.

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36

Kyei, Yaw. "Higher-Order Accurate Finite Volume Discretization of the Three-Dimensional Poisson Equation Based on An Equation Error Method." International Journal for Innovation Education and Research 6, no. 6 (June 30, 2018): 107–23. http://dx.doi.org/10.31686/ijier.vol6.iss6.1076.

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Анотація:

Efficient higher-order accurate finite volume schemes are developed for the threedimensional Poisson’s equation based on optimizations of an equation error expansion on local control volumes. A weighted quadrature of local compact fluxes and the flux integral form of the equation are utilized to formulate the local equation error expansions. Efficient quadrature weights for the schemes are then determined through a minimization of the error expansion for higher-order accurate discretizations of the equation. Consequently, the leading numerical viscosity coefficients are more accurately and completely determined to optimize the weight parameters for uniform higher-order convergence suitable for effective numerical modeling of physical phenomena. Effectiveness of the schemes are evaluated through the solution of the associated eigenvalue problem. Numerical results and analysis of the schemes demonstrate the effectiveness of the methodology.

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37

Filbet, Francis. "Convergence of a Finite Volume Scheme for the Vlasov--Poisson System." SIAM Journal on Numerical Analysis 39, no. 4 (January 2001): 1146–69. http://dx.doi.org/10.1137/s003614290037321x.

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38

Nicaise, Serge. "A posteriori error estimations of some cell-centered finite volume methods." SIAM Journal on Numerical Analysis 43, no. 4 (January 2005): 1481–503. http://dx.doi.org/10.1137/s0036142903437787.

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39

Wendland, Holger. "On the Convergence of a General Class of Finite Volume Methods." SIAM Journal on Numerical Analysis 43, no. 3 (January 2005): 987–1002. http://dx.doi.org/10.1137/040612993.

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40

Cai, Zhiqiang, Jan Mandel, and Steve McCormick. "The Finite Volume Element Method for Diffusion Equations on General Triangulations." SIAM Journal on Numerical Analysis 28, no. 2 (April 1991): 392–402. http://dx.doi.org/10.1137/0728022.

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41

Süli, Endre. "Convergence of Finite Volume Schemes for Poisson’s Equation on Nonuniform Meshes." SIAM Journal on Numerical Analysis 28, no. 5 (October 1991): 1419–30. http://dx.doi.org/10.1137/0728073.

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Maire, P. H., and N. Therme. "Weak Consistency of a Staggered Finite Volume Scheme for Lagrangian Hydrodynamics." SIAM Journal on Numerical Analysis 58, no. 3 (January 2020): 1592–612. http://dx.doi.org/10.1137/19m1259067.

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Chen, Qingshan. "Error analysis of staggered finite difference finite volume schemes on unstructured meshes." Numerical Methods for Partial Differential Equations 33, no. 4 (February 3, 2017): 1159–82. http://dx.doi.org/10.1002/num.22137.

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Handlovičová, A., and Z. Krivá. "PERONA-MALIK EQUATION - ERROR ESTIMATES FOR EXPLICIT FINITE VOLUME SCHEME." Mathematical Modelling and Analysis 10, no. 4 (December 31, 2005): 353–66. http://dx.doi.org/10.3846/13926292.2005.9637293.

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Thomas, J. M., and D. Trujillo. "Mixed finite volume methods." International Journal for Numerical Methods in Engineering 46, no. 9 (November 30, 1999): 1351–66. http://dx.doi.org/10.1002/(sici)1097-0207(19991130)46:9<1351::aid-nme702>3.0.co;2-0.

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46

Kurganov, Alexander. "Finite-volume schemes for shallow-water equations." Acta Numerica 27 (May 1, 2018): 289–351. http://dx.doi.org/10.1017/s0962492918000028.

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Анотація:

Shallow-water equations are widely used to model water flow in rivers, lakes, reservoirs, coastal areas, and other situations in which the water depth is much smaller than the horizontal length scale of motion. The classical shallow-water equations, the Saint-Venant system, were originally proposed about 150 years ago and still are used in a variety of applications. For many practical purposes, it is extremely important to have an accurate, efficient and robust numerical solver for the Saint-Venant system and related models. As their solutions are typically non-smooth and even discontinuous, finite-volume schemes are among the most popular tools. In this paper, we review such schemes and focus on one of the simplest (yet highly accurate and robust) methods: central-upwind schemes. These schemes belong to the family of Godunov-type Riemann-problem-solver-free central schemes, but incorporate some upwinding information about the local speeds of propagation, which helps to reduce an excessive amount of numerical diffusion typically present in classical (staggered) non-oscillatory central schemes. Besides the classical one- and two-dimensional Saint-Venant systems, we will consider the shallow-water equations with friction terms, models with moving bottom topography, the two-layer shallow-water system as well as general non-conservative hyperbolic systems.

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Norton, Richard A., Colin Fox, and Malcolm E. Morrison. "Numerical Approximation of the Frobenius--Perron Operator using the Finite Volume Method." SIAM Journal on Numerical Analysis 56, no. 1 (January 2018): 570–89. http://dx.doi.org/10.1137/16m1108698.

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Michel, Anthony. "A Finite Volume Scheme for Two-Phase Immiscible Flow in Porous Media." SIAM Journal on Numerical Analysis 41, no. 4 (January 2003): 1301–17. http://dx.doi.org/10.1137/s0036142900382739.

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Droniou, Jérôme, Thierry Gallouët, and Raphaèle Herbin. "A Finite Volume Scheme for a Noncoercive Elliptic Equation with Measure Data." SIAM Journal on Numerical Analysis 41, no. 6 (January 2003): 1997–2031. http://dx.doi.org/10.1137/s0036142902405205.

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50

Carstensen, C., R. Lazarov, and S. Tomov. "Explicit and Averaging A Posteriori Error Estimates for Adaptive Finite Volume Methods." SIAM Journal on Numerical Analysis 42, no. 6 (January 2005): 2496–521. http://dx.doi.org/10.1137/s0036142903425422.

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