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Numerical Analysis Of Some Finite Volume Element And Finite Volume Schemes

Posted on:2006-02-12Degree:DoctorType:Dissertation
Country:ChinaCandidate:M YangFull Text:PDF
GTID:1100360155967067Subject:Computational Mathematics
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Finite volume element (FVE) and finite volume (FV) methods have a long history as a class of discretization tools for the numerical simulation of various types of conservation laws. The methods have been widely used in several engineering fields, such as fluid mechanics, heat and mass transfer or petroleum engineering. In the early literature [2-6] they were investigated as the so-called box schemes, generalized difference schemes, and integral finite difference methods, and most of the results were given in linear cases. The Methods are based on a "balance" approach: a local balance is writen on each discretization cell which is often called "control volume" or "dual element"; by the divergence formula, an integral formulation of the equation on the control volume is obtained; then the integral formulation is discretized with respect to the discrete unknowns. Generally speaking, FVE and FV methods are numerical techniques that lie somewhere between finite element and finite difference methods; they have a flexibility similar to that of finite element methods for handling complicated solution domain geometries and boundary conditions; and they have a simplicity for implementation comparable to finite difference methods.The main difference between the FVE and FV methods is that the FVE methods use two spaces: the solution space of piecewise polynomial functions over the primal partition and the test space of piecewise constant functions over the dual partition, and similar as the finite element methods the unknowns are approximated by a Galerkin expansion, while in the FV methods the discrete schemes are obtained directly by use of the finite difference approach to approximate the balance equation on each control volume.The popularity of this class of methods stems from their structural simplicity and the presence of local conservation properties of the numerical fluxes. The second feature makes the methods quite attractive when modelling the problems for which the flux is of importance, such as in fluid mechanics, semi-conductor device simulation, heat and mass transfer ? ? â–  However, the methods also have some serious flaws: 1. essentially the spatial convergence rates of all current FVE and FV schemes are low, only first order in general; 2. for the FVE methods, since the solution space and the test space are different, the coefficient matrices of the discrete schemes are usually not symmetric even for the problems with constant coefficients in multidimensional, which brings much difficulties to analyze or compute them. In fact, many efficient methods used in engineering fields, e.g., the conjugate gradient method, rely on the symmetry of the coefficient matrix.There are two main ways to obtain the high accuracy schemes. 1. One way is to use uniform or symmetric meshes. FVE schemes for elliptic and integro-differential equations on symmetricmeshes with second order spatial convergence in H1-norm have been considered by Cai, Ewing and etc in [8, 9]. FV schemes on triangular meshes were constructed by Vassilevski, Petrova and Lazarov [10]. The error estimates obtained in [10] are in a discrete i^-norm and for uniform triangulations include superconvergence rates, namely, O(h2). FV discretizations on tensor-product nonuniform meshes have been shown by Weiser and Wheeler [11] and superconvergence error estimates were derived. Some upwind FV schemes on square cells, including modified upwind scheme and Il'in's scheme, have been presented by Lazarov, Mishev and Vassilevski [12]. The error estimates in a discrete i^-norm of order O(/im-1), 3/2 < m < 3 were derived there. But this way needs special regular meshes and is only efficient to solve simple cases. Its application is inevitably restricted in practice. 2. For FVE methods, another way to construct high accuracy schemes is to choose high order finite elements as solution space. Based on quadratic Lagrange elements and cubic Hermite elements over triangulations, two kinds of FVE schemes have been first proposed for Poisson equations by Tian and Chen [13, 14] and the optimal if terror estimates were obtained there. Another quadratic scheme has also been presented by Chen [15] for heat conduction equations with constant coefficients, and the optimal energy norm error estimates of order O(At + h2) were derived. For linear elliptic problems, based on biquadratic elements over rectangular meshes, other two FVE schemes have been proposed by Xiang and Feng in [16, 17]. The optimal J?1-norm error estimates of order O(h2) were proved. But until now the second way is also still limited to handle linear problems.Thus, the first objective of this thesis is to generalize the methods about the construction of high accuracy FVE schemes in [13-15] to more general problems. Based on quadratic Lagrange elements and cubic Hermite elements over triangular meshes, we construct and analyze some FVE schemes for nonlinear parabolic systems, nonlinear convection diffusion problems and variant coefficient elliptic problems. We prove the optimal error estimates and give some numerical experiments to demonstrate the feasibility of such generalization.To the best of our knowledge, the literature on the construction of symmetric FVE schemes is little. By the lumped mass methods and approximation replacement of the weak forms, some symmetric schemes for linear elliptic and parabolic problems have been constructed by Liang and Ma in [18, 19], and the error estimates in different norms were derived. Modified symmetric FVE methods both for linear elliptic and parabolic problems have been developed by Rui [20]. The optimal HJ-norm and energy norm error estimates were proved. The solution space considered in these papers is linear Lagrange elements on triangular meshes. Symmetric FVE discretizations for Possion equations, based on biquadratic Lagrange elements, have been considered by Feng in [17]. The application of the numerical integration formulas leads to symmetric schemes and the optimal if^norm error estimates of order O(h2) were obtained. However, all the methods mentioned above are only limited to simple linear problems. They are not feasible to handle more general non-stationary problems.So the second objective of this thesis is to develop a new way to construct symmetric FVEschemes, which can be applied to more general non-stationary problems, such as nonlinear convection diffusion problems. We introduce the alternating direction methods into the construction of the FVE schemes. The choice of this strategy is critical in this paper, which grantees the efficiency and symmetry of the discrete schemes. Moreover, based on biquadratic Lagrange elements, we introduce a new dual partition and by use of such dual partition we can construct high accuracy symmetric FVE schemes even for complicated nonlinear problems.Two-phase incompressible flow is one of the most important fields in petroleum engineering. Generally, the problem is assumed to be positive definite and then various finite element or finite difference methods can be adopted to solve such problem [9, 21-24]. However, in many practical situations the diffusion coefficient is only positive semidefinite, which arises much difficulties in mathematics and mechanics. Some characteristic finite element methods [25, 26] or characteristic finite difference methods [27] have been constructed by Dawson and Yuan to solve such case. But the finite element methods can't keep the local conservation properties of the original problems, while the finite difference methods bring the considerable geometrical error due to the approximation of curved domain with rectangular meshes. On the other hand, the characteristic method needs the spatial interpolation, hence it is very complicated in the practical calculation. For positive definite two-phase incompressible displacement with homogeneous Neuman boundary condition, an upwind single step FV scheme on unstructured meshes has been presented by Michel [28]. Applying compactness theorems, he proved that the scheme is convergent but didn't obtain the specific convergence rate.Therefore, the third objective of this thesis is to construct and study cell-centered FV methods for positive semidefinite two-phase incompressible flow in porous media. We consider more complicated boundary conditions, including inflow and outflow boundaries, A multistep upwind FV scheme on unstructured meshes has been presented for such problem. By use of some numerical techniques, we obtain the specific convergence rate in discrete norms, which is O(At2 + h).This dissertation is divided into four chapters.In Chapter 1, we consider several high accuracy FVE schemes for different problems over triangular meshes. In Section 1.1, we present a fully discrete FVE scheme for a class of nonlinear parabolic systems based on the quadratic Lagrange elements and derive the optimal error estimates in energy norm. Section 1.1.2 contains some necessary results for error estimation. The convergence analysis is presented in Section 1.1.3 and a simple numerical experiment is given in Section 1.1.4.In Section 1.2, we present a FVE scheme with second order accuracy for nonlinear convection diffusion equations. We choose the quadratic Lagrange elements as the solution space and adopt muitistep backward difference methods to increase the accuracy of approximate solution on the temporal direction. In order to reduce numerical diffusion and nonphysical oscillation, we combine the FVE methods with the methods of characteristics. The treatment of the multistep discretization part in FVE methods is different from that in finite element or in finite difference methods. Hence, in the beginning of Section 1.2.2. we put forward an important lemma to solve this difficulty.In Section 1.2.3, we compare the numerical results produced by the multistep scheme with those produced by the single step scheme.In Section 1.3, we generalize the results in [14] for diffusion reaction problems with variant coefficients. We discretize the equation by a cubic FVE method. Several useful Lemmas are given in Section 1.3.2. The numerical results in Section 1.3.4 indicate the efficiency of the scheme.In Chapter 2, we construct two symmetric FVE schemes along characteristics for 2-D and 3-D convection diffusion problems on Tensor product meshes. The solution space considered in this Chapter is the linear Tensor product elements. We introduce the alternating direction methods into the construction of the FVE schemes, which ensures the symmetry and efficiency of the schemes. In Section 2.1, we present a scheme for 3-D convection diffusion problems with nonlinear reaction term. The optimal H1-norm error estimates of order O(At + h) are derived in Section 2.1.3. The results of extensive numerical experiments performed with the alternating direction FVE methods and seven-nodes finite difference methods are given in Section 2.1.4. In Section 2.2, we propose a multistep symmetric scheme for 2-D nonlinear convection diffusion problems. We prove the optimal i?1-norm error estimates of order O(At2 + h) in Section 2.2.3 and compare the numerical results of the single step scheme with those of the multistep one in Section 2.2.4.In Chapter 3, we present a symmetric FVE scheme with second order accuracy for nonlinear convection diffusion problems in 2-D. The equations are discretized by biquadratic FVE methods in space and by multistep methods in time. We introduce a new dual partition to guarantee the symmetry of the both corresponding bilinear form and inner form. Similar as in Chapter 2, the alternating direction strategy is also used in the construction of the scheme. On the basis of such dual partition and the techniques adopted in Chapter 2, we derive the optimal convergence rate of the scheme. In Section 3.2, the primary partition and dual partition are constructed. Various discrete norms used in convergence analysis are also defined. We prove some Lemmas in Section 3.3 and obtain the optimal H!-norm error estimates of order O(At2 + h2) in Section 3.4. In section 3.5, the numerical results produced by the multistep FVE scheme are compared with those of the single step one.In Chapter 4, we propose and study a FV scheme for positive semidefinite problem of two-phase incompressible flow in porous media with initial-boundary conditions in multidimensional. We are interested in some unstructured meshes. Such meshes, including triangular meshes, rectangular meshes, Voronoi meshes and etc, are very popular in practical engineering fields. Upwind strategies are adopted to ensure the stability of the scheme. Multistep methods are used to raise the accuracy on the temporal direction. In section 4.1, we state the problem and make some assumptions on the coefficients. Section 4.2 includes the definition of the unstructured meshes and the discrete FV scheme. In section 4.3, thorough analysis is given to the scheme and the optimal error estimates in discrete norms are proved. Numerical experiments of both positive definite and positive semidefinite cases are given in Section 4.4.
Keywords/Search Tags:finite volume element methods, finite volume methods, high accuracy, symmetric scheme, positivi-semidefinite problem, error estimates
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