Fractional Variational Approach for Dissipative Mechanical Systems

More recently, a variational approach has been proposed by Lin and Wang for damping motion with a Lagrangian holding the energy term dissipated by a friction force. However, the modified Euler-Lagrange equation obtained within their for- malism leads to an incorrect Newtonian equation of motion due to the nonlocality of the Lagrangian. In this communication, we generalize this approach based on the fractional actionlike variational approach and we show that under some simple restric- tions connected to the fractional parameters introduced in the fractional formalism, this problem may be solved.     

A FINITE DIFFERENCE SCHEME FOR SOLVING THE NONLINEAR POISSON-BOLTZMANN EQUATION MODELING CHARGED SPHERES

In this work, we propose an efficient numerical method for computing the electrostaticinteraction between two like-charged spherical particles which is governed by the nonlinearPoisson-Boltzmann equation. The nonlinear problem is solved by a monotone iterativemethod which leads to a sequence of linearized equations. A modified central finite differ-ence scheme is developed to solve the linearized equations on an exterior irregular domainusing a uniform Cartesian grid. With uniform grids, the method is simple, and as aconsequence, multigrid solvers can be employed to speed up the convergence. Numericalexperiments on cases with two isolated spheres and two spheres confined in a chargedcylindrical pore are carried out using the proposed method. Our numerical schemes arefound efficient and the numerical results are found in good agreement with the previouspublished results.     

SCHWARZ ALTERNATING METHOD AND MULTIGRID METHOD

In this paper we propose and establish the convergence of several asynchronous parallel algorithms which are defined by combining the Schwarz alternating method with the multigrid method in two ways. In the first we divide the original problem into p related subproblems, then the multigrid method is used to solve these subproblems iteratively. In the second approach we regard the Schwarz iterative method as the smoothing step of multigrid method. All algorithms proposed in this paper can be used on an MIMD computer.     

THE MORTAR ELEMENT METHOD FOR A NONLINEAR BIHARMONIC EQUATION

The mortar element method is a new domain decomposition method（DDM） with nonoverlapping subdomains. It can handle the situation where the mesh on different subdomains need not align across interfaces, and the matching of discretizations on adjacent subdomains is only enforced weakly. But until now there has been very little work for nonlinear PDEs. In this paper, we will present a mortar-type Morley element method for a nonlinear biharmonic equation which is related to the well-known Navier-Stokes equation. Optimal energy and H^1-norm estimates are obtained under a reasonable elliptic regularity assumption.     

Optimal Control Problem Governed by Semilinear Parabolic Equation and its Algorithm

In this paper, an optimal control problem governed by semilinear parabolic equation which involves the control variable acting on forcing term and coefficients appearing in the higher order derivative terms is formulated and analyzed. The strong variation method, due originally to Mayne et al to solve the optimal control problem of a lumped parameter system, is extended to solve an optimal control problem governed by semilinear parabolic equation, a necessary condition is obtained, the strong variation algorithm for this optimal control problem is presented, and the corresponding convergence result of the algorithm is verified.     

ANTI-DIFFUSIVE FINITE DIFFERENCE WENO METHODS FOR SHALLOW WATER WITH TRANSPORT OF POLLUTANT

In this paper we further explore and apply our recent anti-diffusive flux corrected highorder finite difference WENO schemes for conservation laws [18] to compute the Saint-Venant system of shallow water equations with pollutant propagation, which is describedby a transport equation. The motivation is that the high order anti-diffusive WENOscheme for conservation laws produces sharp resolution of contact discontinuities whilekeeping high order accuracy for the approximation in the smooth region of the solution.The application of the anti-diffusive high order WENO scheme to the Saint-Venant systemof shallow water equations with transport of pollutant achieves high resolution     

Generalized Tricomi Problem for a Quasilinear Mixed Type Equation

In this paper, the Tricomi problem and the generalized Tricomi problem for a quasilinear mixed type equation are studied. The coefficients of the mixed type equation are discontinuous on the line, where the equation changes its type. The existence of solution to these problems is proved. The method developed in this paper can be used to study more difficult problems for nonlinear mixed type equations arising in gas dynamics.     

Energy Decay for the Cauchy Problem of the Linear Wave Equation of Variable Coefficients with Dissipation

Decay of the energy for the Cauchy problem of the wave equation of variable coefficients with a dissipation is considered. It is shown that whether a dissipation can be localized near infinity depends on the curvature properties of a Riemannian metric given by the variable coefficients. In particular, some criteria on curvature of the Riemannian manifold for a dissipation to be localized are given.     

An Improvement of Multigrid Methods Using Multiple Grids on Each Layer for Parallel Computing

Multigrid methods are widely used and well studied for linear solvers and preconditioners of Krylov subspace methods. The multigrid method is one of the most powerful approaches for solving large scale linear systems;however, it may show low parallel efficiency on coarse grids. There are several kinds of research on this issue. In this paper, we intend to overcome this difficulty by proposing a novel multigrid algorithm that has multiple grids on each layer.Numerical results indicate that the proposed method shows a better convergence rate compared with the existing multigrid method.     

On Boundary Stability of Wave Equations with Variable Coefficients

In this paper, we consider the boundary stabilization of the wave equation with variable coefficients by Riemmannian geometry method subject to a different geometric condition which is motivated by the geometric multiplier identities. Several (multiplier) identities (inequalities) which have been built for constant wave equation by Kormornik and Zuazua are generalized to the variable coefficient case by some computational techniques in Riemmannian geometry, so that the precise estimates on the exponential decay rate are derived from those inequalitities. Also, the exponential decay for the solutions of semilinear wave equation with variable coefficients is obtained under natural growth and sign assumptions on the nonlinearity. Our method is rather general and can be adapted to other evolution systems with variable coefficients (e.g. elasticity plates) as well.     

An unconditionally energy stable finite difference scheme for a stochastic Cahn-Hilliard equation

In this work, the MMC-TDGL equation, a stochastic Cahn-Hilliard equation, is solved numerically by using the finite difference method in combination with a convex splitting technique of the energy functional. For the non-stochastic case, we develop an unconditionally energy stable difference scheme which is proved to be uniquely solvable. For the stochastic case, by adopting the same splitting of the energy functional, we construct a similar and uniquely solvable difference scheme with the discretized stochastic term. The resulted schemes are nonlinear and solved by Newton iteration. For the long time simulation, an adaptive time stepping strategy is developed based on both first- and second-order derivatives of the energy. Numerical experiments are carried out to verify the energy stability, the efficiency of the adaptive time stepping and the effect of the stochastic term.     

The statement and numerical solution of an optimization problem in X-ray tomography

A nonclassical problem is considered for the transport equation with coefficients depending on the energy of radiation. The task is to find the discontinuity surfaces for the coefficients of the equation from measurements of the radiation flux leaving the medium. For this tomography problem, an optimization problem is stated and numerically analyzed. The latter consists in determining the radiation energy that ensures the best reconstruction of the unknown medium. A simplified optimization problem is solved analytically.     

A parallel adaptive finite volume method for nanoscale double-gate MOSFETs simulation

We propose in this paper a quantum correction transport model for nanoscale double-gate metal-oxide-semiconductor field effect transistor (MOSFET) device simulation. Based on adaptive finite volume, parallel domain decomposition, monotone iterative, and a posteriori error estimation methods, the model is solved numerically on a PC-based Linux cluster with MPI libraries. Quantum mechanical effect plays an important role in semiconductor nanoscale device simulation. To model this effect, a physical-based quantum correction equation is derived and solved with the hydrodynamic transport model. Numerical calculation of the quantum correction transport model is implemented with the parallel adaptive finite volume method which has recently been proposed by us in deep-submicron semiconductor device simulation. A 20 nm double-gate MOSFET is simulated with the developed quantum transport model and computational technique. Compared with a classical transport model, it is found that this model can account for the quantum mechanical effects of the nanoscale double-gate MOSFET quantitatively. Various biasing conditions have been verified on the simulated device to demonstrate its accuracy. Furthermore, for the same tested problem, the parallel adaptive computation shows very good computational performance in terms of the mesh refinements, the parallel speedup, the load-balancing, and the efficiency.     

A Lions Domain Decomposition Algorithm for Radiation Diffusion Equations on Non-Matching Grids

We develop a Lions domain decomposition algorithm based on a cell functional minimization scheme on non-matching multi-block grids for nonlinear radiation diffusion equations, which are described by the coupled radiation diffusion equations of electron, ion and photon temperatures. The $L^2$orthogonal projection is applied in the Robin transmission condition of non-matching surfaces. Numerical results show that the algorithm keeps the optimal accuracy on the whole computational domain, is robust enough on distorted meshes and curved surfaces, and the convergence rate does not depend on Robin coefficients. It is a practical and attractive algorithm in applying to the two-dimensional three-temperature energy equations of Z-pinch implosion simulation.     

Distributed algebraic multigrid for finite element computations

The Finite Element Method has been successfully applied to a variety of problems in engineering, medicine, biology, and physics. However, this method can be computationally intensive, particularly for problems in which an unstructured mesh of elements is generated. In such situations, the Algebraic Multigrid (AMG) can prove to be a robust method for solving the discretized linear systems that emerge from the problem. Unfortunately, AMG requires a large amount of storage (thus causing swapping on most sequential machines), and typically converges slowly. We show that distributing the algorithm across a cluster of workstations can help alleviate these problems. The distributed algorithm is run on a number of geomechanics problems that are solved using finite elements. The results show that distributed processing is extremely useful in maintaining the performance of the AMG algorithm with increasing problem size, particularly by reducing the amount of disk swapping required.     

用观察法求二阶变系数齐线性方程的非零特解

主要用观察法求二阶变系数齐线性方程的非零特解.首先推广了欧拉方程和另外一个关于求非零特解的结论,其次在已有结果的情形外增添一些可用观察法求特解的情形.     

Time discretization via Laplace transformation of an integro-differential equation of parabolic type

We consider the discretization in time of an inhomogeneous parabolic integro-differential equation, with a memory term of convolution type, in a Banach space setting. The method is based on representing the solution as an integral along a smooth curve in the complex plane which is evaluated to high accuracy by quadrature, using the approach in recent work of López-Fernández and Palencia. This reduces the problem to a finite set of elliptic equations with complex coefficients, which may be solved in parallel. The method is combined with finite element discretization in the spatial variables to yield a fully discrete method. The paper is a further development of earlier work by the authors, which on the one hand treated purely parabolic equations and, on the other, an evolution equation with a positive type memory term. The authors acknowledge the support of the Australian Research Council.     

Boundary value problems for second-order differential operator equations

It is proved that the resolution problem of an operator boundary-value problem for a second-order differential operator equation with constant coefficients is solved in terms of solutions of certain algebraic operator equations. Explicit expressions of solutions are given.     

A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature

We consider the discretization in time of an inhomogeneous parabolicequation in a Banach space setting, using a representation ofthe solution as an integral along a smooth curve in the complexleft half-plane which, after transformation to a finite interval,is then evaluated to high accuracy by a quadrature rule. Thisreduces the problem to a finite set of elliptic equations withcomplex coefficients, which may be solved in parallel. The paperis a further development of earlier work by the authors, wherewe treated the homogeneous equation in a Hilbert space framework.Special attention is given here to the treatment of the forcingterm. The method is combined with finite-element discretizationin spatial variables.     

Formulas that Represent Cauchy Problem Solution for Momentum and Position Schrödinger Equation

In the paper we derive two formulas representing solutions of Cauchy problem for two Schrödinger equations: one-dimensional momentum space equation with polynomial potential, and multidimensional position space equation with locally square integrable potential. The first equation is a constant coefficients particular case of an evolution equation with derivatives of arbitrary high order and variable coefficients that do not change over time, this general equation is solved in the paper. We construct a family of translation operators in the space of square integrable functions and then use methods of functional analysis based on Chernoff product formula to prove that this family approximates the solution-giving semigroup. This leads us to some formulas that express the solution for Cauchy problem in terms of initial condition and coefficients of the equations studied.