Morton, K.W. and Suli, E. (1991). Finite volume methods and their analysis. IMA Journal of Numerical Analysis, 11, 241--60. 70  Home/Search   Document Not in Database   Summary   Related Articles   Check

....The convergence theory of such schemes in several space dimensions has only recently been undertaken. In the case of vertex centered finite volume schemes, studies were carried out by [34] in the case of Cartesian meshes, 18] 3] 5] 6] and [38] in the case of unstructured meshes; see also [28], 36] 24] 29] and [35] in the case of quadrilateral meshes in two space dimensions. Cell centered finite volume schemes are addressed in [25] 15] 41] and [23] in the case of Cartesian meshes and in [40] 19] 20] 22] 26] in the case of triangular (in two space dimensions) or Voronoi ....

Morton, K.W. and E. S uli (1991), Finite volume methods and their analysis, IMA J. Numer. Anal. 11, 241-260.

....modelling flows in aerodynamics. Since the method fits very naturally with convection problems, it has advantageous properties for convection diffusion problems. Nevertheless, all analyses for cell vertex methods have been carried out either for pure convection problems (see, e.g. Morton and Suli [10], Suli [15, 16] and Morton and Stynes [9] or for convection diffusion two point boundary value problems (see, e.g. Mackenzie and Morton [8] and Morton and Stynes [9] So far, there has been no similar analysis for a parabolic convectiondiffusion problem in the literature. In this paper, we ....

....we observe that the cell vertex formulation of the finite volume method has a 3 natural interpretation as a Petrov Galerkin finite element method. The finite element framework then affords the possibility of applying finite element techniques to estimate errors in the finite volume method; see [10, 15, 16, 9]. To reformulate the cell vertex finite volume scheme (2.1) 2.3) as a finite element method, we first define our trial and test spaces. Set U h 0 = Phi v 2 H 1( Omega Gamma C( Omega ) v(0; t) v(1; t) 0 for t 2 (0; T ] v is bilinear on each cell K Psi ; M h = Phi p 2 ....

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K. W. Morton and E. Suli, Finite volume methods and their analysis, IMA J. Numer. Anal. 11 (1991), pp. 241 -- 260.

....numerically by the trapezium rule for the approximation of u and by some suitable difference scheme for the approximation of ru. The cell vertex finite volume method for a pure convective problem (so no approximation for ru is necessary) can be viewed as a Petrov Galerkin finite element method [34]. Petrov Galerkin methods are described later in this chapter. Unlike cell centred schemes the cell vertex method suffers from counting problems in that the number of unknowns will not, in general, match the number of cells (there will be one equation per cell) To overcome this problem either ....

Morton, K. W. and Suli, E. (1991). Finite Volume Methods and their Analysis. IMA J. Numer. Anal. 11, 241-260.

....grids; these grids may consist of polygonal control volumes satisfying adequate geometrical conditions (which are stated in the sequel) and not necessarily ordered in a cartesian grid. We shall be concerned here with the so called cell centered finite volume method. We refer to [1] 17] [24] and references therein for studies on the vertex centered finite volume method, and to [3] 4] 14] and [11] for the related finite volume element and control volume finite element methods. The analysis of cell centered finite volume schemes has only recently been undertaken. Error estimates ....

Morton, K.W. and E. S uli (1991), Finite volume methods and their analysis, IMA J. Numer. Anal. 11, 241-260.

....paper is twofold. i) We like to investigate finite volume discretizations of (2) and wish to establish conditions under which such a discretization implies a corresponding finite volume discretization of the hidden conservation laws (4) and (6) We will focus on a particular cell vertex method [17, 10] which is described in x2. In x3 4, we will derive conditions on the grid structure that implies discrete conservation of symplecticity and momentum. We stress that we only consider smooth solutions of (2) and do not discuss the computation of fronts and shocks. ii) Even when computing smooth ....

....cell) is given by 1 VC Z Omega C ( f(z) j g(z) d dj = 1 VC Z Omega C h(z) d dj which is equivalent to 1 VC I Omega C (f (z) dj Gamma g(z) d ) 1 VC Z Omega C h(z) d dj ; 24) where VC is the oriented area (volume) of the cell. We now use the cell vertex approach [17, 10] to discretize this equation. The unknowns z i are held at the cell vertices ( i ; j i ) i = 1; 4. See Fig. 1, where the independent variables are denoted by (x; y) The midpoint rule is used along the edges to calculate the fluxes. 3 To do so we introduce the midpoint approximations ....

K. W. Morton & E. S uli, Finite volume methods and their analysis, IMA J. Num. Anal. 11 (1991), 241--260.

.... finite volume difference methods on triangular meshes are well defined and monotone under quite restrictive conditions (acuteangled triangles [37, 15] or constant coefficients [13] Interesting results have been reported for quadrilateral vertex centered finite volume difference schemes [25, 22, 33, 31]. but the consistent theory for such meshes is still not available. We derive two schemes, central difference scheme (CDS) for symmetric or diffusion dominated problems and upwind difference scheme (UDS) for convection dominated problems and show that they are stable and first order accurate. ....

K. W. Morton and E. Suli. Finite volume methods and their analysis. IMA J.Numer. Anal., 11:241--260, 1991.

....here by the so called cell centered approach, i.e. the discrete unknowns are located at some point in the control volumes. For the finite volume element and control volume finite element approaches where the unknowns are located at the vertices, see [5] 4] 15] 16] and [10] see also [25] for other types of finite volume methods. Error estimates and convergence results for cell centered finite volume schemes for the discretization of linear elliptic equations have recently been studied. The one dimensional case and multi dimensional rectangular cases were studied in [23] 17] ....

Morton, K.W. and E. S uli (1991), Finite volume methods and their analysis, IMA J. Numer. Anal. 11, 241-260.

....The convergence theory of such schemes in several space dimensions has only recently been undertaken. In the case of vertex centered finite volume schemes, studies were carried out by [41] in the case of Cartesian meshes, 25] 3] 5] 6] and [45] in the case of unstructured meshes; see also [36], 43] and [33] in the case of quadrilateral meshes. We are interested here by the so called cell centered approach, i.e. the discrete unknowns are located at some point in the control volumes. Cell centered finite volume schemes are addressed in [34] 23] 50] and [30] in the case of ....

Morton, K.W. and E. S uli (1991), Finite volume methods and their analysis, IMA J. Numer. Anal. 11, 241-260.

.... by modern FV error analysis for the lowest order methods based on FE style arguments for linear finite elements: e.g. Bank and Rose [5] Ewing et al. 17, 18] Hackbusch [21] Heinrich [23] Herbin [24] Kreiss et al. 31] Lazarov et al. 32, 33] Manteuffel and White [35] Morton and Suli [38], Samarskii et al. 46] Suli [50] and Weiser and Wheeler [54] By adapting FV and FE arguments to fit the context of FVE and developing some new arguments unique to FVE, we derive optimal order error estimates under the full range of admissible regularities for the exact solution for h, p, and ....

....(see Mitchell and Griffiths [36] Kreiss et al. 31] Manteuffel and White [35] and the references listed therein) FV is viewed as a special type of finite difference method and analyzed in terms of finite difference truncation error in C 1 norms. However, in modern finite volume analysis (see [5, 17, 18, 21, 23, 32, 33, 38, 46, 50, 54]) FV is viewed as a special type of finite element method and its error is analyzed in terms of finite element approximation theory results: results for the error in approximating elements of a fractional order Sobolev space with piecewise polynomials. Since FVE is even closer to FE than FV, our ....

K. W. Morton and E. Suli, "Finite volume methods and their analysis," IMA J. Numer. Anal., 11 (1991), pp. 241--260.

....2 ) Cell centered discretizations on tensor product nonuniform meshes were considered by Weiser and Wheeler [25] and superconvergence error estimates were derived. H 1 error estimates of order O(h 1 ff ) 1 2 ff 1 for the Poisson equation were proved by S uli [23] Morton and S uli [17] considered point centered finite difference schemes for one and two dimensional hyperbolic equations. A method closely related to the finite element approximations is the finite volume element method proposed and analyzed by Cai [6] Cai, Mandel and McCormick [7] and McCormick [16] see also an ....

K. W. Morton and E. Suli, Finite volume methods and their analysis, IMA J. Numer. Anal., 11 (1991), pp. 241--260.

....uniformly on Omega Gamma Clearly (b ) is sufficient for (b) when d = 2. Under this assumption U h Gamma and M h have the same dimension and L h is then easily seen to be bijective. For theoretical results concerning the stability and convergence of the cell vertex scheme we refer to [2] [18], 19] 22] 23] and [24] Here we derive an a posteriori error bound for the method. Theorem 3.4 Suppose that hypothesis (b ) holds, that F = fT h g is a family of structured quasi parallel partitions of Omega , and that, given s, 1 s 3, the coefficients of the matrices A i , i = 1; 2, ....

Morton, K.W. and S¨uli, E. (1991). Finite volume methods and their analysis. IMA Journal of Numerical Analysis, 11, 241--60. 33

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Morton, K.W. and Suli, E. (1991). Finite volume methods and their analysis. IMA Journal of Numerical Analysis, 11, 241--60. 70

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Morton, K.W. and E. S uli (1991), Finite volume methods and their analysis, IMA J. Numer. Anal. 11, 241-260.

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K.W. Morton and E. S uli (1991), Finite volume methods and their analysis, IMA J. Numer. Anal. 11, 241-260.

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K. W. Morton and E. S uli, Finite volume methods and their analysis. IMA J. Numer. Anal., Vol. 11, (1991), pp. 241--260.

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K.W. Morton and E. S uli (1991), Finite volume methods and their analysis, IMA J. Numer. Anal. 11, 241-260.

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K.W. Morton and E. S uli (1991), Finite volume methods and their analysis, IMA J. Numer. Anal. 11, 241-260.

No context found.

K. W. Morton and E. S uli, Finite volume methods and their analysis. IMA J. Numer. Anal., Vol. 11, (1991), pp. 241--260.

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