著作名稱：  SEMIEXACT SOLUTIONS AND PULSATING FRONTS FOR LOTKAVOLTERRA SYSTEMS OF TWO COMPETING SPECIES IN SPATIALLY PERIODIC HABITATS 
年度：  2020 
類別： 
期刊論文
COMMUNICATIONS ON PURE AND APPLIED ANALYSIS

摘要：  We are concerned with the coexistence states of the diffusive LotkaVolterra system of two competing species when the growth rates of the two species depend periodically on the spacial variable. For the onedimensional problem, we employ the generalized Jacobi elliptic function method to find semiexact solutions under certain conditions on the parameters. In addition, we use the sine function to construct a pair of upper and lower solutions and obtain a solution of the abovementioned system. Next, we provide a sufficient condition for the existence of pulsating fronts connecting two semitrivial states by applying the abstract theory regarding monotone semiflows. Some numerical simulations are also included. 
關鍵字：  Semiexact solutions;traveling wave solutions;reactiondiffusion equations 
著作名稱：  Positive steady states of reaction–diffusion–advection competition models in periodic environment 
年度：  2017 
類別： 
期刊論文
Journal of Mathematical Analysis and Applications

摘要：  In this paper, we consider the positive steady states for reaction–diffusion–advection competition models in the whole space with a spatially periodic structure. Under the spatially periodic setting, we establish sufficient conditions for the existence of positive steady states of this model, respectively, by investigating the sign of the principal eigenvalue for some linearized eigenvalue problems. As an application, a Lotka–Volterra reaction–diffusion–advection model for two competing species in a spatially periodic environment is considered. Finally, some numerical simulations are presented to seek dynamical behaviors. 
關鍵字：  Positive steady states, Reaction–diffusion–advection, Population dynamics, Periodic environment 
著作名稱：  A Null Space Free JacobiDavidson Iteration for Maxwells Operator 
年度：  2015 
類別： 
期刊論文
SIAM Journal on Scientific Computing

摘要：  We present an efficient null space free Jacobi–Davidson method to compute the
positive eigenvalues of time harmonic Maxwell’s equations. We focus on a class of spatial discretizations that guarantee the existence of discrete vector potentials, such as Yee’s scheme and the edge
elements. During the Jacobi–Davidson iteration, the correction process is applied to the vector potential instead. The correction equation is solved approximately as in the standard Jacobi–Davidson
approach. The computational cost of the transformation from the vector potential to the corrector is
negligible. As a consequence, the expanding subspace automatically stays out of the null space and
no extra projection step is needed. Numerical evidence confirms that the proposed scheme indeed
outperforms the standard and projectionbased Jacobi–Davidson methods by a significant margin. 
關鍵字：  time harmonic Maxwell’s equations, Yee’s scheme, edge elements, generalized eigenvalue problem, discrete vector potential, discrete deRham complex, Poincar’e Lemma, Jacobi– Davidson method 
著作名稱：  A GENERALIZED MAC SCHEME ON CURVILINEAR DOMAINS 
年度：  2013 
類別： 
期刊論文
SIAM Journal on Scientific Computing

摘要：  We propose a simple finite difference scheme for Navier–Stokes equations in primitive formulation on curvilinear domains. With proper boundary treatment and interplay between covariant and contravariant components, the spatial discretization admits exact Hodge decomposition and energy identity. As a result, the pressure can be decoupled from the momentum equation with explicit time stepping. No artificial pressure boundary condition is needed. In addition, it can be shown that this spatially compatible discretization leads to uniform infsup condition, which plays a crucial role in the pressure approximation of both dynamic and steady state calculations. Numerical experiments demonstrate the robustness and efficiency of our scheme. 
關鍵字：  Navier–Stokes equations, infsup condition, MarkAndCell scheme, incompressible flow, pressure boundary condition, pressure Poisson equation 
著作名稱：  Preconditioning bandgap eigenvalue problems in threeDimensional photonic crystals simulations 
年度：  2010 
類別： 
期刊論文
Journal of Computational Physics

摘要：  To explore band structures of threedimensional photonic crystals numerically, we need to solve the eigenvalue problems derived from the governing Maxwell equations. The solutions of these eigenvalue problems cannot be computed effectively unless a suitable combination of eigenvalue solver and preconditioner is chosen. Taking eigenvalue problems due to Yees scheme as examples, we propose using Krylov–Schur method and JacobiDavidson method to solve the resulting eigenvalue problems. For preconditioning, we derive several novel preconditioning schemes based on various preconditioners, including a preconditioner that can be solved by Fast Fourier Transform efficiently. We then conduct intensive numerical experiments for various combinations of eigenvalue solvers and preconditioning schemes. We find that the KrylovSchur method associated with the Fast Fourier Transform based preconditioner is very efficient. It remarkably outperforms all other eigenvalue solvers with common preconditioners like Jacobi, Symmetric Successive Over Relaxation, and incomplete factorizations. This promising solver can benefit applications like photonic crystal structure optimization. 
關鍵字：  Threedimensional photonic crystals, Maxwells equations, Eigenvalue problems, Preconditioning, Fast Fourier transform, KrylovSchur method, JacobiDavidson method, Harmonic extraction 
著作名稱：  An FFT Based Fast Poisson Solver on Spherical Shells 
年度：  2011 
類別： 
期刊論文
Communications in Computational Physic

摘要：  We present a fast Poisson solver on spherical shells. With a special change of variable, the radial part of the Laplacian transforms to a constant coefficient differential
operator. As a result, the Fast Fourier Transform can be applied to solve the Poisson equation with O(N^3 log N) operations. Numerical examples have confirmed the accuracy and robustness of the new scheme. 
關鍵字：  Poisson equation, spherical coordinate, FFT, spectralfinite difference method, fast diagonalization, high order accuracy, error estimate, trapezoidal rule, EulerMaclaurin formula, Bernoulli numbers. 