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DISCRETE ENERGY ANALYSIS OF THE THIRD-ORDER VARIABLE-STEP BDF TIME-STEPPING FOR Diffusion equations

Journal of Computational Mathematics 2023年第2期41卷 325-344页

作者： Hong-lin Liao Tao Tang Tao Zhou Department of Mathematics Nanjing University of Aeronautics and AstronauticsNanjing 211106China Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles(NUAA) MIITNanjing 211106China Division of Science and Technology BNU-HKBU United International CollegeZhuhai 519087China Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing100190China

This is one of our series works on discrete energy analysis of the variable-step BDF schemes.In this part,we present stability and convergence analysis of the third-order BDF(BDF3)schemes with variable steps for linear Diffusion equations,see,e.g.,[SIAM J.Numer.Anal.,58:2294-2314]and[Math.Comp.,90:1207-1226]for our previous works on the BDF2 scheme.To this aim,we first build up a discrete gradient structure of the variable-step BDF3 formula under the condition that the adjacent step ratios are less than 1.4877,by which we can establish a discrete energy dissipation law.Mesh-robust stability and convergence analysis in the L^(2) norm are then obtained.Here the mesh robustness means that the solution errors are well controlled by the maximum time-step size but independent of the adjacent time-step ratios.We also present numerical tests to support our theoretical results.

关键词： Diffusion equations Variable-step third-order BDF scheme Discrete gradient structure Discrete orthogonal convolution kernels Stability and convergence

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The Corrected Finite Volume Element Methods for Diffusion equations Satisfying Discrete Extremum Principle

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Communications in Computational Physics 2022年第10期32卷 1437-1473页

作者： Ang Li Hongtao Yang Yonghai Li Guangwei Yuan School of Mathematics Jilin UniversityChangchun 130012JilinChina Institute of Applied Physics and Computational Mathematics Fenghaodong RoadHaidian DistrictBeijing 100094China

In this paper,we correct the finite volume element methods for Diffusion equations on general triangular and quadrilateral meshes.First,we decompose the numerical fluxes of original schemes into two parts,i.e.,the principal part with a twopoint flux structure and the defective part.And then with the help of local extremums,we transform the original numerical fluxes into nonlinear numerical fluxes,which can be expressed as a nonlinear combination of two-point fluxes.It is proved that the corrected schemes satisfy the discrete strong extremum principle without restrictions on the diffusion coefficient and meshes.Numerical results indicate that the corrected schemes not only satisfy the discrete strong extremum principle but also preserve the convergence order of the original finite volume element methods.

关键词： Diffusion equations finite volume element flux-correct maximum principle

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Extension of the High-Order Space-Time Discontinuous Galerkin Cell Vertex Scheme to Solve Time Dependent Diffusion equations

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Communications in Computational Physics 2012年第5期11卷 1503-1524页

作者： Shuangzhang Tu Gordon W.Skelton Qing Pang Department of Computer Engineering Jackson State UniversityJacksonMS 39217USA

In this paper,the high-order space-time discontinuous Galerkin cell vertex scheme(DG-CVS)developed by the authors for hyperbolic conservation laws is extended for time dependent Diffusion equations.In the extension,the treatment of the diffusive flux is exactly the same as that for the advective flux.Thanks to the Riemannsolver-free and reconstruction-free features of DG-CVS,both the advective flux and the diffusive flux are evaluated using continuous information across the cell interface.As a result,the resulting formulation with diffusive fluxes present is still consistent and does not need any extra ad hoc techniques to cure the common“variational crime”problem when traditional DG methods are applied to diffusion problems.For this reason,DG-CVS is conceptually simpler than other existing DG-typed methods.The numerical tests demonstrate that the convergence order based on the L_(2)-norm is optimal,i.e.O(h^(p+1))for the solution and O(h^(p))for the solution gradients,when the basis polynomials are of odd degrees.For even-degree polynomials,the convergence order is sub-optimal for the solution and optimal for the solution gradients.The same odd-even behaviour can also be seen in some other DG-typed methods.

关键词： High-order method space-time method discontinuous Galerkin(DG)method cellvertex scheme(CVS) Diffusion equations

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EXISTENCE OF SOLUTIONS FOR GRADIENT SYSTEMS WITH APPLICATION TO diffusion PROBLEMS INVOLVING NONCONVEX ENERGIES

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Acta Mathematica Scientia 2018年第6期38卷 1767-1778页

作者： Sahbi BOUSSANDEL Dpartement de Mathmatiques Facult des Sciences de Bizerte

In this paper, we establish the existence of solutions for gradient systems ofevolution under some type （M） and semi-coerciveness conditions. The main result is appliedin order to solve nonlinear Diffusion equations... 详细信息

关键词： gradient systems existence of solutions Galerkin method type （M） condition semi-coerciveness Diffusion equations

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