Dong Liang

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• Professor
• Professor (Full) at York University

This paper presents a mini immersed finite element (IFE) method for solving two- and three-dimensional two-phase Stokes problems on Cartesian meshes. The IFE space is constructed from the conventional mini element, with shape functions modified on interface elements according to interface jump conditions while keeping the degrees of freedom unchang...

This paper presents a mini immersed finite element (IFE) method for solving two- and three-dimensional two-phase Stokes problems on Cartesian meshes. The IFE space is constructed from the conventional mini element with shape functions modified on interface elements according to interface jump conditions, while keeping the degrees of freedom unchang...

In this paper, an efficient modified method of characteristics with adjust advection (MMOCAA) domain decomposition method (DDM) is analyzed for solving advection diffusion equations. On each non-overlapping block divided sub-domains, we take two half time steps along x-direction and y-direction, to solve the numerical solutions on each sub-domain a...

In this paper, the energy-conserved splitting multidomain Legendre-tau Chebyshev-collocation spectral method is proposed for solving two dimensional Maxwell’s equations. The method uses different degree polynomials to approximate the electric and magnetic fields respectively, and they can be decoupled in computation. Moreover, the error estimates a...

Nonlinear contaminant transports through porous media are important in many scientific and engineering applications. In this paper, we develop and analyze fourth-order block-centered compact difference scheme (BCCDS) for the nonlinear contaminant transport equations with adsorption process in porous media. Based on block-centered mesh, a fourth ord...

In this paper, energy-preserving time high-order average vector field (AVF) compact finite difference scheme is proposed and analyzed for solving two-dimensional nonlinear wave equations including the nonlinear sine-Gordon equation and the nonlinear Klein-Gordon equation. We first present the corresponding Hamiltonian system to the two-dimensional...

Based on the combination of block-centered and compact difference methods, fourth order compact block-centered finite difference schemes for the numerical solutions of one-dimensional and two-dimensional elliptic and parabolic problems with variable coefficients are derived and analyzed. Stability and optimal fourth-order error estimates are proved...

In this paper, we develop and analyze a new time fourth-order energy-preserving average vector field (AVF) finite difference method for the nonlinear fractional wave equations with Riesz space-fractional derivative. To the corresponding Hamiltonian system of the nonlinear fractional wave equations, the fourth-order weighted and shifted Lubich diffe...

Interface partial differential equations (PDEs) are very important in science and engineering. A new mass preserving solution-flux scheme is proposed in this paper for solving parabolic multi-layer interface problems. In the scheme, the domain is divided into staggered meshes for layers. At grid points in each subdomain, the solution-flux scheme is...

In this paper, we develop and analyze two types of new energy-preserving local mesh-refined splitting finite difference time-domain (EP-LMR-S-FDTD) schemes for two-dimensional Maxwell's equations. For the local mesh refinements, it is challenging to define the suitable local interface schemes which can preserve energy and guarantee high accuracy. T...

Eddy currents are induced by rapidly switching currents of the gradient coils in a MRI scanner, which is typically cylindrical in shape, hence the use of cylindrical co-ordinates is more numerically efficient. We develop the energy conserved splitting finite different time domain (EC-S-FDTD) scheme in the cylindrical co-ordinates to estimate the ed...

In this article, we develop and analyze two energy-preserving high-order average vector field (AVF) compact finite difference schemes for solving variable coefficient nonlinear wave equations with periodic boundary conditions. Specifically, we first consider the variable coefficient nonlinear wave equation as an infinite-dimensional Hamiltonian sys...

In this paper, two new efficient energy dissipative difference schemes for the strongly coupled nonlinear damped space fractional wave equations are first set forth and analyzed, which involve a two-level nonlinear difference scheme, and a three-level linear difference scheme based on invariant energy quadratization formulation. Then the discrete e...

The coupled sine-Gordon (SG) equations and the coupled Klein-Gordon (KG) equations play an important role in scientific fields, such as nonlinear optics, solid state physics, quantum mechanics. As their energies are conservative, it is of importance to develop energy preserving finite difference method (EP-FDM) for these systems of nonlinear wave e...

We consider a numerical scheme for incompressible miscible displacement problem in porous media. A multipoint flux mixed finite element method is used to handle the velocity–pressure equation. The standard finite element method is used to approximate the concentration equation. Error estimates for pressure and velocity and concentration are present...

Aerosol particles have an important effect on changing of climate and human health, where aerosols scatter and absorb the incoming solar radiation, and thus decrease the precipitation efficiency of warm clouds and can cause an indirect radiative forcing associated with changes in cloud properties. Meanwhile, it has also been recognized that the par...

The surface fluid flows coupled with porous media flows in substrates occur in many circumstances in industry and natural settings. In this paper, we investigate the long wave solutions for the surface flows on inclined porous media. The important feature is that such flows are derived by the Navier-Stokes equations governing the clear flows in the...

In this paper, we give the energy conservation for the electromagnetic fields propagating in two-dimensional Lorentz media and develop a new spatial fourth-order compact splitting FDTD scheme to solve the two-dimensional electromagnetic Lorentz system. The spatial compact finite difference technique and the splitting technique are combined to const...

In the paper, a new conservative splitting decomposition method (S-DDM) is developed for computing nonlinear multicomponent contamination flows in porous media over multi-block sub-domains. On each block-divided sub-domain, we take three steps to solve the coupled nonlinear system of water-head equation and multicomponent concentration equations in...

In this paper, we develop and analyze the energy conservative time high-order AVF compact finite difference methods for variable coefficient acoustic wave equations in two dimensions. We first derive out an infinite-dimensional Hamiltonian system for the variable coefficient wave equations and apply the spatial fourth-order compact finite differenc...

In this paper, we develop the immersed finite element method for parabolic optimal control problems with interfaces. By employing the definition of directional derivative of Lagrange function, first-order necessary optimality conditions in qualified form for parabolic optimal control problems with interfaces are established. The parabolic state equ...

In this paper, we develop a new energy conservative splitting FDTD scheme for solving the metamaterial electromagnetic Lorentz system. The electromagnetic Lorentz model in metamaterials is to describe the resonance of nuclei-bounded electrons in dielectrics by the damped oscillation with a restoring force. Investigating the energy properties for me...

In this paper, a new energy dissipative fourth-order difference scheme for the high-dimensional nonlinear fractional generalized wave equations is constructed. Then, the discrete energy dissipation property of the system is exhibited in detail. Next, we prove that the proposed scheme is uniquely solvable. By the discrete energy method, it is shown...

In this paper, an efficient energy-preserving splitting Crank-Nicolson difference scheme for the fractional-in-space Boussinesq equation is established. The novelty of the proposed scheme here is that the potential function v is introduced via [Formula presented] to ensure the conservation of energy. By utilizing the discrete energy method, it is s...

In this paper, we propose and analyze a conservative high-order compact finite difference scheme for the Klein–Gordon–Schrödinger (KGS) equation with Dirichlet boundary condition. By introducing the time-derivative of one solution as a separate dependent variable, we rewrite the KGS equation as a system of partial differential equations (PDEs) and...

This paper develops and analyzes the optimal weighted upwind finite volume method (OWUFVM) for convection–diffusion equations on nonuniform rectangular meshes. The dual meshes in this method are associated with the local Peclet numbers, which are determined by the convection and diffusion coefficients and the mesh size. We first prove the stability...

In this paper, two conservative and fourth-order compact finite difference schemes are proposed and analyzed for solving the regularized long wave (RLW) equation. The first compact finite difference scheme is two-level and nonlinear implicit. The second scheme is three-level and linearized implicit. Conservations of the discrete mass and energy, an...

We develop a second-order corrected-explicit–implicit domain decomposition scheme (SCEIDD) for the parallel approximation of convection–diffusion equations over multi-block subdomains. The stability and convergence properties of the SCEIDD scheme are analyzed. It is proved that the scheme is unconditionally stable and it is second-order accurate in...

In this paper, a new time second-order characteristic finite element method is proposed and analyzed for solving the advection-diffusion equations involving a nonlinear right side term. In order to obtain a time second-order characteristic scheme, the global derivative term transferred from the time derivative and advection terms is discretized by...

In the article, a new and efficient mass-conserving operator splitting domain decomposition method (S-DDM) is proposed and analyzed for solving two dimensional variable coefficient parabolic equations with reaction term. The domain is divided into multiple non-overlapping block-divided subdomains. On each block-divided subdomain, the interface flux...

In this paper, by applying order reduction approach, a second-order accurate box scheme is established to solve a nonlinear delayed convection-diffusion equations with Neumann boundary conditions. By the discrete energy method, it is shown that the difference scheme is uniquely solvable, and has a convergence rate of O(Δt2+h2) with respect to L2- n...

This article proposes a class of high-order energy-preserving schemes for the improved Boussinesq equation. To derive the energy-preserving schemes, we first discretize the improved Boussinesq equation by Fourier pseudospectral method, which leads to a finite-dimensional Hamiltonian system. Then, the obtained semidiscrete system is solved by Hamilt...

In this paper, we propose and analyze a finite difference method for the nonlinear Schrödinger equations on unbounded domain by using artificial boundary conditions. Two artificial boundary conditions are introduced to restrict the original Schrödinger equations on an unbounded domain into an initial–boundary value problem with a bounded domain. Th...

In this paper, the new mass-preserving time second-order explicit–implicit domain decomposition (DD) schemes for solving parabolic equations with various coefficients are proposed. In the schemes, first, the interface values on the interfaces of sub-domains are explicitly calculated, in which two kinds of explicit difference schemes are used. Secon...

We develop a multipoint flux mixed finite element method for the compressible Darcy–Forchheimer models. This method is motivated by the multipoint flux approximation method. Based on the lowest order Brezzi–Douglas–Marini mixed finite element method and combined with a special quadrature rule that allows for local velocity elimination and leads to...

We propose in this paper a time second order mass conservative algorithm for solving advection–diffusion equations. A conservative interpolation and a continuous discrete flux are coupled to the characteristic finite difference method, which enables using large time step size in computation. The advection–diffusion equations are first transformed t...

In this paper, a compact alternating direction implicit (ADI) method, which combines the fourth-order compact difference for the approximations of the second spatial derivatives and the approximation factorizations of difference operators, is firstly presented for solving two-dimensional (2D) second order dual-phase-lagging models of microscale hea...

In this paper, we focus on constructing and analysing a new Fourier pseudo-spectral conservative scheme for the Klein–Gordon–Schrödinger (KGS) equation. After rewriting the KGS equation as an infinite-dimensional Hamiltonian system, we use a Fourier pseudo-spectral method to discrete the system in space to obtain a semi-discrete system, which can b...

In the paper, a new time second-order mass-conserved implicit/explicit domain decomposition method (DDM) for the diffusion equations is proposed. In the scheme, firstly, we calculate the interface fluxes of sub-domains from the obtained solutions and fluxes at the previous time level, for which we apply high-order Taylor’s expansion and transfer th...

In this paper, a new high-order energy-preserving scheme is proposed for the modified Korteweg-de Vries equation. The proposed scheme is constructed by using the Hamiltonian boundary value methods in time, and Fourier pseudospectral method in space. Exploiting this method, we get second-order and fourth-order energy-preserving integrators. The prop...

In this article, we study the Drude models of Maxwell's equations in three-dimensional metamaterials. We derive new global energy-tracking identities for the three dimensional electromagnetic problems in the Drude metamaterials, which describe the invariance of global electromagnetic energy in variation forms. We propose the time second-order globa...

In this paper, we develop a new groundwater pollution control by finding the optimal amounts of fluxes for disposals, where the optimization objective function is proposed by considering the overall effects caused by pollution, abatement costs and pollution taxes. The proposed optimal control model is subject to the flow and transport groundwater e...

In this paper, a new conservative and splitting fourth-order compact difference scheme is proposed and analyzed for solving two-dimensional linear Schrodinger equations. The proposed splitting high-order compact scheme in two dimensions has the excellent property that it preserves the conservations of charge and energy. We strictly prove that the s...

In the paper, a new mass-preserving and modified-upwind S-DDM scheme over non-overlapping multi-block sub-domains for solving time-dependent convection-diffusion equations is developed and analyzed. On each sub-domain, the intermediate fluxes on the interfaces of sub-domains are firstly computed by the modified semi-implicit flux schemes. Then, the...

A new time second-order symmetric energy-conserved splitting FDTD scheme is proposed for solving the two-dimensional Maxwell’s equations in negative index metamaterials with Drude model. The scheme is proved to preserve the discrete electromagnetic energies in metamaterials and is of second order accuracy both in time and space. The proposed scheme...

In this paper, a splitting characteristic method is developed for solving general multi-component aerosol transports in atmosphere, which can efficiently compute the aerosol transports by using large time step sizes. The proposed characteristic finite difference method (C-FDM) can solve the multi-component aerosol distributions in high dimensional...

Aerosol modeling is very important to study the behavior of aerosol dynamics in atmospheric environment. In this paper we consider numerical methods for the nonlinear aerosol dynamic equations on time and particle size. The finite volume element methods based on the linear interpolation and Hermite interpolation are provided to approximate the aero...

In the paper, we develop and analyze a new mass-preserving splitting domain decomposition method over multiple sub-domains for solving parabolic equations. The domain is divided into non-overlapping multi-bock sub-domains. On the interfaces of sub-domains, the interface fluxes are computed by the semi-implicit (explicit) flux scheme. The solutions...

An efficient and accurate numerical scheme is proposed for solving the transverse electric (TE) mode electromagnetic (EM) propagation problem in two-dimensional earth. The scheme is based on the alternating direction finite-difference time-domain (ADI-FDTD) method. Unlike the conventional upward continuation approach for the earth-air interface, an...

In this work, we develop and analyze mathematical models for the coupled within-host and between-host dynamics caricaturing the evolution of HIV/AIDS. The host population is divided into susceptible, the infected without receiving treatment and the infected receiving ART treatment in accordance with China’s Four-Free-One-Care Policy. The within-hos...

In the paper, we develop the new conservative characteristic finite difference methods (C-CFD) for the atmospheric aerosol transport problems. We propose the time second-order and spatial high-order conservative characteristic finite difference methods for the aerosol vertical advection-diffusion process and the two-dimensional conservative charact...

In this paper, we analyze the energy-conserved splitting finite-difference time-domain (FDTD) scheme for variable coefficient Maxwell's equations in two-dimensional disk domains. The approach is energy-conserved, unconditionally stable, and effective. We strictly prove that the EC-S-FDTD scheme for the variable coefficient Maxwell's equations in di...

An efficient time second-order characteristic finite element method for solving the nonlinear multi-component aerosol dynamic equations is developed. While a highly accurate characteristic method is used to treat the advection multi-component condensation/evaporation process, a time high-order extrapolation along the characteristics is applied to a...

In this paper, we develop and analyze the efficient splitting domain decomposition method for solving parabolic equations. The domain is divided into non-overlapping multi-block subdomains. On interfaces of sub-domains, the local multilevel explicit scheme is proposed to solve the interface values of solution, and then the splitting implicit scheme...

In this paper, the energy-conserved splitting Legendre Galerkin method and energy-conserved splitting Legendre collocation method for Maxwell’s equations in two dimensions are proposed. The schemes are energy-conserved, unconditionally stable, and can be implemented efficiently. The both methods are of second-order convergence in time. The high ord...

In the paper two new locally one-dimensional alternating segment schemes for solving the 2-D parabolic problems are developed. The parabolic problems can be solved efficiently over decomposed sub-domains. Two schemes are proved to be unconditionally stable and numerical experiments show the validity of the schemes.

Aerosol modeling is very important to study the behavior of aerosol dynamics in atmospheric environment. In this paper we consider the nonlinear aerosol dynamic equations on time and particle size, which involve the advection–condensation process and the nonlinear coagulation process. For solving accurately the multiple sharp log-normal aerosol dis...

In modern numerical simulation of problems in energy resources and environmental science, it is important to develop efficient numerical methods for time‐dependent convection–diffusion problems. On the basis of nonstandard covolume grids, we propose a new kind of high‐order upwind finite volume element method for the problems. We first prove the st...

In this paper we develop a new spatial fourth-order energy-conserved splitting finite-difference time-domain method for Maxwell’s equations. Based on the staggered grids, the splitting technique is applied to lead to a three-stage energy-conserved splitting scheme. At each stage, using the spatial fourth-order difference operators on the strict int...

We analyze an adaptive wavelet method with variable time step sizes and space refinement for parabolic equations. The advantages of multi-resolution wavelet processes combined with certain equivalences involving weighted sequence norms of wavelet coefficients allow us to set up an efficient adaptive algorithm producing locally refined spaces for ea...

We develop a new energy-conserved S-FDTD scheme for the Maxwell’s equations in metamaterials. We first derive out the new property of energy conservation of the governing equations in metamaterials, and then propose the energy-conserved S-FDTD scheme for solving the problems based on the staggered grids. We prove that the proposed scheme is energy-...

In this paper, a multi-functional moving-cut high-dimensional model representation (MC-HDMR) approach is developed for simulation of multi-component input and output aerosols. This method leads to an aerosol prediction database system based on full thermodynamic models such as ISORROPIA. The developed prediction system can efficiently compute the p...

The symmetric energy-conserved splitting FDTD scheme developed in [1] is a very new and efficient scheme for computing theMaxwell's equations. It is based on splitting the whole Maxwell's equations and matching the x-direction and y-direction electric fields associated to the magnetic field symmetrically. In this paper, we make further study on the...

For the numerical simulation of seawater intrusion and the consequences of protection projects, a modular form of project adjustment in porous media, and the modified upwind finite difference, fractional steps schemes are put forward. Based on the numerical simulation of a practical situation in the Laizhou Bay Area of Shandodng Province, predictiv...

We present a mathematical model parameterized to simulate the 1918 pandemic and modified to account for today's achievements in medical care and technology. Our goal is to use the model with carefully selected parameters to analyze and simulate different scenarios in a changing environment including behavior changes and reduction of mobility as the...

In the paper, we develop a new method of combining the singular limit argument and the singular perturbation technique to establish the existence of the point-to-periodic heteroclinic travelling wave solutions connecting an equilibrium and a periodic travelling wave solution for a delayed predator–prey diffusion partial differential equation (PDE)...

The nonlinear version of the mixed spectral finite difference model of atmospheric boundary-layer flow over topography is reviewed. The relations between the stability of the iteration scheme and its relaxation parameter are discussed. Suitable choice of the relaxation factor improves the computational stability on terrain with maximum slope up to...

In this paper, we analyze two new second-order characteristic schemes in time and age for an age-structured population model with nonlinear diffusion and reaction. By using the characteristic difference to approximate the transport term and the average along the characteristics to treat the nonlinear spatial diffusion and reaction terms, an implici...

The high dimensional model representation (HDMR) method was recently proposed as an efficient tool to capture the input-output relationships in high-dimensional systems for many problems in science and engineering. In this paper, we develop a new multiple sub-domain random sampling HDMR method (MSD-RS-HDMR) for general high dimensional input-output...

In this paper, we develop an efficient splitting domain decomposition method (S-DDM) for compressible contamination fluid flows in porous media over multiple block-divided sub-domains by combining the non-overlapping domain decomposition, splitting, linearization and extrapolation techniques. The proposed S-DDM iterative approach divides the large...

In this paper, we develop a new and efficient approach for high dimensional atmospheric aerosol thermodynamic equilibrium predictions. The multi-phase and multi-component aerosol thermodynamic input–output systems are solved by the high dimensional model representation (HDMR) method combining with the moving multiple cut points. The developed appro...

In this paper, we develop and analyze efficient energy-conserved splitting finitedifference time-domain (FDTD) schemes for solving three dimensional Maxwell's equations in electromagnetic computations. All proposed energy-conserved splitting finite-difference time-domain (EC-S-FDTD) algorithms are strictly proved to be energy-conserved and uncondit...

In the paper we consider the non-linear aerosol dynamic equation on time and particle size, which contains the advection process of condensation growth and the process of non-linear coagulation. We develop an efficient second-order characteristic finite element method for solving the problem. A high accurate characteristic method is proposed to tre...

In this paper, a new symmetric energy-conserved splitting FDTD scheme (symmetric EC-S-FDTD) for Maxwell's equations is proposed. The new algorithm inherits the same properties of our previous EC-S-FDTDI and EC-S-FDTDII algorithms: energy-conservation, unconditional stability and computational efficiency. It keeps the same computational complexity a...

Aerosol modelling is very important to study and simulate the behavior of aerosol dynamics in atmospheric environment. In this paper, we consider the gen-eral nonlinear aerosol dynamic equations which describe the evolution of the aerosol distribution. Continuous time and discrete time wavelet Galerkin methods are pro-posed for solving this problem...

In this article we consider the age structured population growth model of marine invertebrates. The problem is a nonlinear coupled system of the age-density distribution of sessile adults and the abundance of larvae. We propose the semidiscrete and fully-discrete discontinuous Galerkin schemes to the nonlinear problem. The DG method is well suited...

In this paper, we develop a new and efficient splitting domain decomposition method for solving parabolic-type time-dependent equations in porous media. The method combines the multi-block non-overlapping domain decomposition and the splitting technique. On interfaces of sub-domains, the local multilevel explicit scheme is proposed to solve the int...

For the three-dimensional seawater intrusion and protection system, the model of dynamics of fluids in porous media and the modified upwind finite difference fractional steps schemes are put forward. Based on the numerical simulation of the practical situation in the Laizhou Bay Area of Shandong Province, predictive numerical simulation and analysi...

In this paper, we derive a population model for the growth of a single species on a two-dimensional strip with Neumann and Robin boundary conditions. We show that the dynamics of the mature population is governed by a reaction–diffusion equation with delayed global interaction. Using the theory of asymptotic speed of spread and monotone traveling w...

In this paper, we perform a systematic multiscale analysis for convection dominated transport equations with a weak diffusion and a highly oscillatory velocity field. The paper primarily focuses on upscaling linear transport equations. But we also discuss briefly how to upscale two-phase miscible flows, in which case the concentration equation is c...

In this paper, we study splitting numerical methods for the three-dimensional Maxwell equations in the time domain. We propose a new kind of splitting finite-difference time-domain schemes on a staggered grid, which consists of only two stages for each time step. It is proved by the energy method that the splitting scheme is unconditionally stable...

In this paper, a new and robust splitting wavelet method has been developed to solve the general aerosol dynamics equation. The considered models are the nonlinear integro-partial differential equations on time, size and space, which describe different processes of atmospheric aerosols including condensation, nucleation, coagulation, deposition, so...

In this paper, two new energy-conserved splitting methods (EC-S-FDTDI and EC-S-FDTDII) for Maxwell’s equations in two dimensions are proposed. Both algorithms are energy-conserved, unconditionally stable and can be computed efficiently. The convergence results are analyzed based on the energy method, which show that the EC-S-FDTDI scheme is of firs...

In this paper, we consider splitting methods for Maxwell's equations in two dimensions. A new kind of splitting finite-difference time-domain methods on a staggered grid is developed. The corresponding schemes consist of only two stages for each time step, which are very simple in computation. The rigorous analysis of the schemes is given. By the e...

In this paper we propose a derivative-free optimization algorithm based on conditional moments for finding the maximizer of an objective function. The proposed algorithm does not require calculation or approximation of any order derivative of the objective function. The step size in iteration is determined adaptively according to the local geometri...