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Monotone Finite Volume Methods For Diffusion Equation On Distorted Meshes

Posted on:2012-10-14Degree:DoctorType:Dissertation
Country:ChinaCandidate:S WangFull Text:PDF
GTID:1110330368978711Subject:Computational Mathematics
Abstract/Summary:PDF Full Text Request
Radiation hydrodynamics is very important in the area of inertial confinement fusion (ICF), magnetic confinement fusion, geology and astrophysics. In these areas, accurate and reliable discretization methods for the diffusion equations on distorted meshes are very important.An accurate and reliable numerical method must mimic fundamental properties of practical problems. The monotonicity and local mass conservation are the essential properties of the diffusion equation. In the context of anisotropic thermal conduction, a discrete scheme without monotone can lead to the violation of the entropy constraints of the second law of thermodynamics, causing heat to flow from regions of lower tem-perature to higher temperature. In regions of large temperature variations, this can cause the temperature to become negative.The outline of this paper is as follows:●The diamond scheme [22-25] is an important scheme for solving diffusion equation. However, the monotonicity of the diamond scheme has not been researched yet. Liska proposed a repair technique [1] for linear (bilinear) finite element solution, and enforced the discrete maximum principle. In Section 2, we extend the repair technique in [1], and propose two postprocess method. Both of them keep total energy conservation, and are easy to be implemented in existing codes. Numerical examples show that these two repair techniques do not destroy the accuracy of solution for the diamond schemes on distorted meshes.●Method in [15] extend the method in [10] to advection-diffusion problems. How-ever, it imposes some restrictions on mesh and it cannot be directly applied to diffusion problems with discontinuous coefficient. In Section 3, we propose a new monotone finite volume method with second-order accuracy for the steady-state advection-diffusion equation. The new method does not need any mesh restric-tions, deal with discontinuous rigorous, monotone. The second-order convergence rate for concentration and the monotonicity of the nonlinear finite volume method are verified with numerical experiments.●Finite volume element method, as finite element method, do not satisfy discrete maximum principle for anisotropic problems on distorted mesh. It derives severe restrictions on diffusion coefficient and on the geometric properties of the mesh. [16] proposed a nonlinear constrained finite volume element method to anisotropic diffusion problems. Compared with a stand (linear or bilinear) finite element method, the entries of the stiffness matrix are adjusted so as to enforce sufficient conditions of discrete maximum principle. In Section 4, we extend the method in [16] to finite volume method. It is suitable for triangular or quadrilateral mesh, also suitable for anisotropic problems and satisfying discrete maximum principle.●In Section 3, we have proposed a monotone finite volume methods for advection-diffusion equations, but its does not satisfy the discrete maximum principle. [20] proposed a finite volume scheme satisfying extreme principle for diffusion. In Section 5, we extend the method [20] to advection-diffusion equation. Similar with interpolation free method, the new scheme has the less interpolation computations.
Keywords/Search Tags:finite volume method, finite volume element method, monotonicity, ex-treme principle
PDF Full Text Request
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