An inverse problem of reconstructing the initial condition for a time fractional diffusion equation is investigated.On the basis of the optimal control framework,the uniqueness and first order necessary optimality condition of the minimizer for the objective functional are established,and a time-space spectral method is proposed to numerically solve the resulting minimization problem.The contribution of the paper is threefold:1)a priori error estimate for the spectral approximation is derived;2)a conjugate gradient optimization algorithm is designed to efficiently solve the inverse problem;3)some numerical experiments are carried out to show that the proposed method is capable to find out the optimal initial condition,and that the convergence rate of the method is exponential if the optimal initial condition is smooth.
In this paper, we introduce a multi-material arbitrary Lagrangian and Eulerian method for the hydrodynamic radiative multi-group diffusion model in 2D cylindrical coordinates. The basic idea in the construction of the method is the following： In the Lagrangian step, a closure model of radiation-hydrodynamics is used to give the states of equations for materials in mixed cells. In the mesh rezoning step, we couple the rezoning principle with the Lagrangian interface tracking method and an Eulerian interface capturing scheme to compute interfaces sharply according to their deformation and to keep cells in good geometric quality. In the interface reconstruction step, a dual-material Moment-of-Fluid method is introduced to obtain the unique interface in mixed ceils. In the remapping step, a conservative remapping algorithm of conserved quantities is presented. A munber of numerical tests are carried out and the numerical results show that the new method can simulate instabilities in complex fluid field under large deformation, and are accurate and robust.
Aiming at the isoparametric bilinear finite volume element scheme,we initially derive an asymptotic expansion and a high accuracy combination formula of the derivatives in the sense of pointwise by employing the energy-embedded method on uniform grids.Furthermore,we prove that the approximate derivatives are convergent of order two.Finally,numerical examples verify the theoretical results.
In present paper,the locomotion of an oblate jellyfish is numerically investigated by using a momentum exchange-based immersed boundary-Lattice Boltzmann method based on a dynamic model describing the oblate jellyfish.The present investigation is agreed fairly well with the previous experimental works.The Reynolds number and the mass density of the jellyfish are found to have significant effects on the locomotion of the oblate jellyfish.Increasing Reynolds number,the motion frequency of the jellyfish becomes slow due to the reduced work done for the pulsations,and decreases and increases before and after the mass density ratio of the jellyfish to the carried fluid is 0.1.The total work increases rapidly at small mass density ratios and slowly increases to a constant value at large mass density ratio.Moreover,as mass density ratio increases,the maximum forward velocity significantly reduces in the contraction stage,while the minimum forward velocity increases in the relaxation stage.
Hai-Zhuan YuanShi ShuXiao-Dong NiuMingjun LiYang Hu
We study the boundary value problem of a coupled differential system of fractional order, and prove the existence and uniqueness of solutions to the considered problem. The underlying differential system is featured by a fractional differential operator, which is defined in the Riemann-Liouville sense, and a nonlinear term in which different solution components are coupled. The analysis is based on the reduction of the given system to an equivalent system of integral equations. By means of the nonlinear alternative of Leray-Schauder,the existence of solutions of the factional differential system is obtained. The uniqueness is established by using the Banach contraction principle.
In this paper,we propose an iterative two-grid method for the edge finite element discretizations(a saddle-point system)of Perfectly Matched Layer(PML)equations to the Maxwell scattering problem in two dimensions.Firstly,we use a fine space to solve a discrete saddle-point system of H(grad)variational problems,denoted by auxiliary system 1.Secondly,we use a coarse space to solve the original saddle-point system.Then,we use a fine space again to solve a discrete H(curl)-elliptic variational problems,denoted by auxiliary system 2.Furthermore,we develop a regularization diagonal block preconditioner for auxiliary system 1 and use H-X preconditioner for auxiliary system 2.Hence we essentially transform the original problem in a fine space to a corresponding(but much smaller)problem on a coarse space,due to the fact that the above two preconditioners are efficient and stable.Compared with some existing iterative methods for solving saddle-point systems,such as PMinres,numerical experiments show the competitive performance of our iterative two-grid method.
Chunmei LiuShi ShuYunqing HuangLiuqiang ZhongJunxian Wang
