AN ARTIFICIAL BOUNDARY CONDITION FOR THE INCOMPRESSIBLE VISCOUS FLOWS IN A NO-SLIP CHANNEL

1.IntroductionWhencomputingthenumericals0luti0nsofviscousfluidfl0wproblemsinallun-boundedd0main,0neoftenintroducesartificialboundaries,andsetsupanartificialbopundarycondition0nthem;thenthe0riginalproblemisreducedtoaproblemonab0undedc0mputationald0main.InordertoIimitthecomputatio11alcost,theseboundariesmustnotbet00farfromthedomainofinterest.Theref0re,theartificialboundaryc0nditi0nsmustbegoodapprotimationt0the"exact"boundaryconditions(sothatthes0lutionoftheproblemintheboundeddonlainisequaltothes…     

Convergence and stability of a numerical method for micromagnetics

The convergence and stability of a numerical method, which applies a nonconforming finite element method and an artificial boundary method to a multi-atomic Young measure relaxation model, for micromagnetics are analyzed. By revealing some key properties of the solution sets of both the continuous and discrete problems, we show that our numerical method is stable, and the solution set of the continuous problem is well approximated by those of the discrete problems. The performance of our method is also illustrated by some numerical examples. The research was supported in part by the Major State Basic Research Projects (2005CB321701), NSFC projects (10431050, 10571006, 10528102 and 10871011) and RFDP of China.     

Deformation Driven Homogenization of Fracturing Solids

The paper discusses numerical formulations of the homogenization for solids with discrete crack development. We focus on multi–phase microstructures of heterogeneous materials, where fracture occurs in the form of debonding mechanisms as well as matrix cracking. The definition of overall properties critically depends on the developing discontinuities. To this end, we extend continuous formulations [1] to microstructures with discontinuities [2]. The basic underlying structure is a canonical variational formulation in the fully nonlinear range based on incremental energy minimization. We develop algorithms for numerical homogenization of fracturing solids in a deformation–driven context with non–trivial formulations of boundary conditions for (i) linear deformation and (ii) uniform tractions. The overall response of composite materials with fracturing microstructures are investigated. As a key result, we show the significance of the proposed non–trivial formulation of a traction–type boundary condition in the deformation–driven context. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Discrete artificial boundary conditions for nonlinear Schrödinger equations

In this work we construct and analyze discrete artificial boundary conditions (ABCs) for different finite difference schemes to solve nonlinear Schrödinger equations. These new discrete boundary conditions are motivated by the continuous ABCs recently obtained by the potential strategy of Szeftel. Since these new nonlinear ABCs are based on the discrete ABCs for the linear problem we first review the well-known results for the linear Schrödinger equation. We present our approach for a couple of finite difference schemes, including the Crank–Nicholson scheme, the Dùran–Sanz-Serna scheme, the DuFort–Frankel method and several split-step (fractional-step) methods such as the Lie splitting, the Strang splitting and the relaxation scheme of Besse. Finally, several numerical tests illustrate the accuracy and stability of our new discrete approach for the considered finite difference schemes.     

Nonlocal Boundary Conditions for Higher–Order Parabolic Equations

This work deals with the efficient numerical solution of the two–dimensional one–way Helmholtz equation posed on an unbounded domain. In this case one has to introduce artificial boundary conditions to confine the computational domain. Here we construct with the Z –transformation so–called discrete transparent boundary conditions for higher–order parabolic equations schemes. These methods are Padé “Parabolic” approximations of the one–way Helmholtz equation and frequently used in integrated optics and (underwater) acoustics. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of a pseudostress-based mixed finite element method for the Brinkman model of porous media flow

In this paper we introduce and analyze a new mixed finite element method for the two-dimensional Brinkman model of porous media flow with mixed boundary conditions. We use a dual-mixed formulation in which the main unknown is given by the pseudostress. The original velocity and pressure unknowns are easily recovered through a simple postprocessing. In addition, since the Neumann boundary condition becomes essential, we impose it in a weak sense, which yields the introduction of the trace of the fluid velocity over the Neumann boundary as the associated Lagrange multiplier. We apply the Babu?ka–Brezzi theory to establish sufficient conditions for the well-posedness of the resulting continuous and discrete formulations. In particular, a feasible choice of finite element subspaces is given by Raviart–Thomas elements of order $k \ge 0$ for the pseudostress, and continuous piecewise polynomials of degree $k + 1$ for the Lagrange multiplier. We also derive a reliable and efficient residual-based a posteriori error estimator for this problem. Suitable auxiliary problems, the continuous inf-sup conditions satisfied by the bilinear forms involved, a discrete Helmholtz decomposition, and the local approximation properties of the Raviart–Thomas and Clément interpolation operators are the main tools for proving the reliability. Then, Helmholtz’s decomposition, inverse inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are employed to show the efficiency. Finally, several numerical results illustrating the performance and the robustness of the method, confirming the theoretical properties of the estimator, and showing the behaviour of the associated adaptive algorithm, are provided.     

Decoupled energy‐law preserving numerical schemes for the Cahn–Hilliard–Darcy system

We study two novel decoupled energy‐law preserving and mass‐conservative numerical schemes for solving the Cahn‐Hilliard‐Darcy system which models two‐phase flow in porous medium or in a Hele–Shaw cell. In the first scheme, the velocity in the Cahn–Hilliard equation is treated explicitly so that the Darcy equation is completely decoupled from the Cahn–Hilliard equation. In the second scheme, an intermediate velocity is used in the Cahn–Hilliard equation which allows for the decoupling. We show that the first scheme preserves a discrete energy law with a time‐step constraint, while the second scheme satisfies an energy law without any constraint and is unconditionally stable. Ample numerical experiments are performed to gauge the efficiency and robustness of our scheme. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 936–954, 2016     

THE GLOBAL ARTIFICIAL BOUNDARY CONDITIONS FOR NUMERICAL SIMULATIONS OF THE 3D FLOW AROUND A SUBMERGED BODY

We consider the numerical approximations of the three-dimensional steady potential flow around a body moving in a liquid of finite constant depth at constant speed and distance below a free surface in a channel. One vertical side is introduced as the up-stream artificial boundary and two vertical sides are introduced as the downstream arti-ficial boundaries. On the artificial boundaries, a sequence of high-order global artificial boundary conditions are given. Then the original problem is reduced to a problem defined on a finite computational domain, which is equivalent to a variational problem. After solving the variational problem by the finite element method, we obtain the numerical approximation of the original problem. The numerical examples show that the artificial boundary conditions given in this paper are very effective.     

Cahn-Hilliard方程的拟谱逼近

该文讨论用Ｌｅｇｅｎｄｒｅ拟谱方法数值求解非线性Ｃａｈｎ Ｈｉｌｌｉａｒｄ方程的Ｄｉｒｉｃｈｌｅｔ问题．建立了其半离散和全离散逼近格式，它们保持原问题能量耗散的性质．证明了离散解的存在唯一性，并给出了最佳误差估计．数值实验也证实了我们的结果．     

THE ARTIFICIAL BOUNDARY CONDITIONS FOR NUMERICAL SIMULATIONS OF THE COMPLEX AMPLITUDE IN A COUPLED BAY-RIVER SYSTEM

We consider the numerical approximations of the complex amplitude in a coupled bayriver system in this work. One half-circumference is introduced as the artificial boundary in the open sea, and one segment is introduced as the artificial boundary in the river if the river is semi-infinite. On the artificial boundary a sequence of high-order artificial boundary conditions are proposed. Then the original problem is solved in a finite computational domain, which is equivalent to a variational problem. The numerical approximations for the original problem are obtained by solving the variational probiem with the finite element method. The numerical examples show that the artificial boundary conditions given in this work are very effective.     

A NON-OVERLAPPING DOMAIN DECOMPOSITION ALGORITHM BASED ON THE NATURAL BOUNDARY REDUCTION FOR WAVE EQUATIONS IN AN UNBOUNDED DOMAIN

In this paper, a new domain decomposition method based on the natural boundary reduction, which solves wave problems over an unbounded domain, is suggestted. An circular artificial boundary is introduced. The original unbounded domain is divided into two subdomains, an internal bounded region and external unbounded region outside the artificial boundary. A Dirichlet-Neumann(D-N) alternating iteration algorithm is constructed. We prove that the algorithm is equavilent to preconditional Richardson iteration method. Numerical studies are performed by finite element method. The numerical results show that the convergence rate of the discrete D-N iteration is independent of the finite element mesh size.     

Numerical methods for weak solution of wave equation with van der Pol type nonlinear boundary conditions

We develop computational methods for solving wave equation with van der Pol type nonlinear boundary conditions under the framework of weak solutions. Based on the wave reflection on the boundaries, we first solve the Riemann invariants by constructing two iteration mappings, and then show that the weak solution can be obtained by the integration of the Riemann invariants on the boundaries. If the compatible conditions are not satisfied or only hold with a low degree, a high‐order integration method is developed for the numerical solution. When the initial condition is sufficiently smooth and compatible conditions hold with a sufficient degree, we establish a sixth‐order finite difference scheme, which only needs to solve a linear system at any given time instance. Numerical experiments are provided to demonstrate the effectiveness of the proposed approaches. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 373–398, 2016     

EXACT NONREFLECTING BOUNDARY CONDITIONS FOR AN ACOUSTIC PROBLEM IN THREE DIMENSIONS

In this paper,nonreflecting artificial boundary,conditions are considered for an acoustic problem in three dimensions.With the technique of Fourier dexompostition under the orthogonal basis of spherical harmonics,three kinds of equivalemt exact artificial boundary conditions are obtained on a spherical artificail boundary.A numerical test is presented to show the performance of the method.     

Cahn—Hilliard方程的Legendre谱逼近

1.引 言本文我们将考虑非线性Cahn—Hilliard方程的初边值问题     

Numerical methods for the linear Boltzmann transport equation in slab geometry

An iterative method for computing numerical solutions of a finite-difference system corresponding to the linear Boltzmann equation in slab geometry is presented. This iterative scheme gives a straightforward marching process starting from the given boundary and initial conditions. It is shown that with a suitable initial iteration the sequence of iterations converges monotonically to a unique solution of the finite-difference system. This monotone convergence leads to improved upper and lower bounds of the solution in each iteration, and to the well-posedness of the discrete system in the sense of Hadamard. It also leads to the convergence of the discrete system to the continuous system as the mesh size of the space–velocity–time variables approaches to zero. Under a mild restriction on the time-increment the discrete system is numerically stable, independent of the mesh-size of the space and velocity. An error estimate for the computed solution due to simultaneous initial and iteration error is obtained. Also given are some numerical results for the time-dependent and the steady-state solutions.     

Pinning sampled‐data synchronization of complex dynamical networks with Markovian jumping and mixed delays using multiple integral approach

This study deals with the pinning synchronization problem for complex dynamical networks (CDNs) with Markovian jumping parameters and mixed delays under sampled‐data control technique. The mixed delays cover both discrete and distributed delays. The Markovian jumping parameters are modeled as a continuous‐time, finite‐state Markov chain. The sufficient conditions for asymptotic synchronization of considered networks are obtained by utilizing novel Lyapunov‐Krasovskii functional and multiple integral approach. The obtained criteria is formulated in terms of LMIs, which can be checked for feasibility by making use of available softwares. Lastly, numerical simulation results are presented to validate the advantage of the propound theoretical results. © 2016 Wiley Periodicals, Inc. Complexity 21: 622–632, 2016     

A simple and robust boundary treatment for the forced Korteweg–de Vries equation

In this paper, we propose a simple and robust numerical method for the forced Korteweg–de Vries (fKdV) equation which models free surface waves of an incompressible and inviscid fluid flow over a bump. The fKdV equation is defined in an infinite domain. However, to solve the equation numerically we must truncate the infinite domain to a bounded domain by introducing an artificial boundary and imposing boundary conditions there. Due to unsuitable artificial boundary conditions, most wave propagation problems have numerical difficulties (e.g., the truncated computational domain must be large enough or the numerical simulation must be terminated before the wave approaches the artificial boundary for the quality of the numerical solution). To solve this boundary problem, we develop an absorbing non-reflecting boundary treatment which uses outward wave velocity. The basic idea of the proposing algorithm is that we first calculate an outward wave velocity from the solutions at the previous and present time steps and then we obtain a solution at the next time step on the artificial boundary by moving the solution at the present time step with the velocity. And then we update solutions at the next time step inside the domain using the calculated solution on the artificial boundary. Numerical experiments with various initial conditions for the KdV and fKdV equations are presented to illustrate the accuracy and efficiency of our method.     

Heat transfer at high energy devices with prescribed cooling flow

We study the heat transfer from a high‐energy electric device into a surrounding cooling flow. We analyse several simplifications of the model to allow an easier numerical treatment. First, the flow variables velocity and pressure are assumed to be independent from the temperature which allows a reduction to Prandtl's boundary layer model and leads to a coupled nonlinear transmission problem for the temperature distribution. Second, a further simplification using a Kirchhoff transform leads to a coupled Laplace equation with nonlinear boundary conditions. We analyse existence and uniqueness of both the continuous and discrete systems. Finally, we provide some numerical results for a simple two‐dimensional model problem. Copyright © 2006 John Wiley & Sons, Ltd.     

Kinetic approximation of a boundary value problem for conservation laws

Summary. We design numerical schemes for systems of conservation laws with boundary conditions. These schemes are based on relaxation approximations taking the form of discrete BGK models with kinetic boundary conditions. The resulting schemes are Riemann solver free and easily extendable to higher order in time or in space. For scalar equations convergence is proved. We show numerical examples, including solutions of Euler equations.Mathematics Subject Classification (2000): 65M06, 65M12, 76M20Correspondence to: D. Aregba-Driollet     

A finite‐volume scheme for a spinorial matrix drift‐diffusion model for semiconductors

An implicit Euler finite‐volume scheme for a spinorial matrix drift‐diffusion model for semiconductors is analyzed. The model consists of strongly coupled parabolic equations for the electron density matrix or, alternatively, of weakly coupled equations for the charge and spin‐vector densities, coupled to the Poisson equation for the electric potential. The equations are solved in a bounded domain with mixed Dirichlet–Neumann boundary conditions. The charge and spin‐vector fluxes are approximated by a Scharfetter–Gummel discretization. The main features of the numerical scheme are the preservation of nonnegativity and bounds of the densities and the dissipation of the discrete free energy. The existence of a bounded discrete solution and the monotonicity of the discrete free energy are proved. For undoped semiconductor materials, the numerical scheme is unconditionally stable. The fundamental ideas are reformulations using spin‐up and spin‐down densities and certain projections of the spin‐vector density, free energy estimates, and a discrete Moser iteration. Furthermore, numerical simulations of a simple ferromagnetic‐layer field‐effect transistor in two space dimensions are presented. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 819–846, 2016