scholarly journals On 3D DDFV Discretization of Gradient and Divergence Operators: Discrete Functional Analysis Tools and Applications to Degenerate Parabolic Problems

2013 ◽  
Vol 13 (4) ◽  
pp. 369-410 ◽  
Author(s):  
Boris Andreianov ◽  
Mostafa Bendahmane ◽  
Florence Hubert
Keyword(s):  
Functional Analysis ◽  
Finite Volume ◽  
Parabolic Problems ◽  
Sobolev Embedding ◽  
Finite Volume Schemes ◽  
Compactness Result ◽  
Nonlinear Elliptic ◽  
Analysis Tools ◽  
Consistency Results ◽  
Discrete Compactness

Abstract. We present a detailed survey of discrete functional analysis tools (consistency results, Poincaré and Sobolev embedding inequalities, discrete W1,p compactness, discrete compactness in space and in time) for the so-called Discrete Duality Finite Volume (DDFV) schemes in three space dimensions. We concentrate mainly on the 3D CeVe-DDFV scheme presented in [IMA J. Numer. Anal., 32 (2012), pp. 1574–1603]. Some of our results are new, such as a general time-compactness result based upon the idea of Kruzhkov (1969); others generalize the ideas known for the 2D DDFV schemes or for traditional two-point-flux finite volume schemes. We illustrate the use of these tools by studying convergence of discretizations of nonlinear elliptic-parabolic problems of Leray–Lions kind, and provide numerical results for this example.

scholarly journals Mixed and Hybrid Finite Element Methods for Convection-Diffusion Problems and Their Relationships with Finite Volume: The Multi-Dimensional Case

2017 ◽  
Vol 9 (1) ◽  
pp. 68 ◽  
Author(s):  
Michel Fortin ◽  
Abdellatif Serghini Mounim
Keyword(s):  
Finite Volume ◽  
Numerical Approximation ◽  
Dimensional Case ◽  
Parabolic Problems ◽  
Finite Volume Schemes ◽  
One Dimensional ◽  
Convection Diffusion ◽  
Hybrid Finite Element ◽  
The One ◽  
Diffusion Problems

We introduced in (Fortin & Serghini Mounim, 2005) a new method which allows us to extend the connection between the finite volume and dual mixed hybrid (DMH) methods to advection-diffusion problems in the one-dimensional case. In the present work we propose to extend the results of (Fortin & Serghini Mounim, 2005) to multidimensional hyperbolic and parabolic problems. The numerical approximation is achieved using the Raviart-Thomas (Raviart & Thomas, 1977) finite elements of lowest degree on triangular or rectangular partitions. We show the link with numerous finite volume schemes by use of appropriate numerical integrations. This will permit a better understanding of these finite volume schemes and the large number of DMH results available could carry out their analysis in a unified fashion. Furthermore, a stabilized method is proposed. We end with some discussion on possible extensions of our schemes.

Topside Defect Localization Using OBIRCH Analysis

2005 ◽  
Author(s):  
Clifford Howard ◽  
Anusha Weerakoon ◽  
Diana M. Mitro ◽  
Dawn Glaeser
Keyword(s):  
Functional Analysis ◽  
Defect Location ◽  
Analysis Tools ◽  
Defect Localization ◽  
Tools And Techniques

Abstract OBIRCH analysis is a useful technique for defect localization not only for parametric failures, but also for functional analysis. However, OBIRCH results do not always identify the exact defect location. OBIRCH analysis results must be used in conjunction with other analysis tools and techniques to successfully identify defect locations.

Transport Schemes in GungHo

2021 ◽  
Author(s):  
James Kent
Keyword(s):  
Finite Element ◽  
Finite Volume ◽  
Mixed Finite Element ◽  
Method Of Lines ◽  
Finite Volume Methods ◽  
Kutta Method ◽  
Finite Volume Schemes ◽  
Dynamical Core ◽  
Runge Kutta Method ◽  
The Stability

<p>GungHo is the mixed finite-element dynamical core under development by the Met Office. A key component of the dynamical core is the transport scheme, which advects density, temperature, moisture, and the winds, throughout the atmosphere. Transport in GungHo is performed by finite-volume methods, to ensure conservation of certain quantaties. There are a range of different finite-volume schemes being considered for transport, including the Runge-Kutta/method-of-lines and COSMIC/Lin-Rood schemes. Additional horizontal/vertical splitting approaches are also under consideration, to improve the stability aspects of the model. Here we discuss these transport options and present results from the GungHo framework, featuring both prescribed velocity advection tests and full dry dynamical core tests. </p>

scholarly journals A Review of Residual Distribution Schemes for Hyperbolic and Parabolic Problems: The July 2010 State of the Art

2012 ◽  
Vol 11 (4) ◽  
pp. 1043-1080 ◽  
Author(s):  
Remi Abgrall
Keyword(s):  
Finite Element ◽  
Discontinuous Galerkin ◽  
State Of The Art ◽  
Natural Extension ◽  
Parabolic Problems ◽  
Residual Distribution ◽  
Finite Volume Schemes ◽  
Stabilized Finite Element ◽  
Research Perspectives ◽  
Residual Distribution Schemes

AbstractWe describe and review non oscillatory residual distribution schemes that are rather natural extension of high order finite volume schemes when a special emphasis is put on the structure of the computational stencil. We provide their connections with standard stabilized finite element and discontinuous Galerkin schemes, show that their are really non oscillatory. We also discuss the extension to these methods to parabolic problems. We also draw some research perspectives.

MOMENT PRESERVING FINITE VOLUME SCHEMES FOR SOLVING POPULATION BALANCE EQUATIONS INCORPORATING AGGREGATION, BREAKAGE, GROWTH AND SOURCE TERMS

2013 ◽  
Vol 23 (07) ◽  
pp. 1235-1273 ◽  
Author(s):  
RAJESH KUMAR ◽  
JITENDRA KUMAR ◽  
GERALD WARNECKE
Keyword(s):  
Finite Volume ◽  
Population Balance ◽  
Coupled Processes ◽  
Source Terms ◽  
Balance Equations ◽  
Finite Volume Schemes ◽  
Average Technique ◽  
Population Balance Equations ◽  
Moment Preservation ◽  
First Moment

In this work we present some moment preserving finite volume schemes (FVS) for solving population balance equations. We are considering unified numerical methods to simultaneous aggregation, breakage, growth and source terms, e.g. for nucleation. The criteria for the preservation of different moments are given. The property of conservation is a special case of preservation. First we present a FVS which shows the preservation with respect to one-moment depending upon the processes under consideration. In case of the aggregation and breakage it satisfies first-moment preservation whereas for the growth and nucleation we observe zeroth-moment preservation. This is due to the well-known property of conservativity of FVS. However, coupling of all the processes shows no preservation for any moments. To overcome this issue, we reformulate the cell average technique into a conservative formulation which is coupled together with a modified upwind scheme to give moment preservation with respect to the first two-moments for all four processes under consideration. This allows for easy coupling of these processes. The preservation is proven mathematically and verified numerically. The numerical results for the first two-moments are verified for various coupled processes where analytical solutions are available.

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