椭圆-抛物偏微分方程奇异摄动问题的差分解法

本文讨论了含有小参数在高阶导数项的椭圆型方程奇异摄动问题的差分解法.当ε=0时椭圆型方程退化为抛物型方程.作者根据此问题解的边界层性质,构造了特殊的差分格式:研究了它的收敛性和解的渐近性态.最后给出一个数值例题.     

Parameter estimation in moving boundary problems

We describe an approximation scheme which can be used to estimate unknown parameters in moving boundary problems. The model equations we consider are fairly general nonlinear diffusion/reaction equations of one spatial variable. Here we give conditions on the parameter sets and model equations under which we can prove that the estimates obtained using the approximations will converge to best-fit parameters for the original model equations. We conclude with a numerical example.     

半线性常微分方程混合边值奇摄动问题的一致差分格式*

本文应用分离奇性法研究半线性常微分方程混合边值奇摄动问题的一致差分格式,我们证明了所构造的I1'in型差分格式关于小参数ε的一阶一致收敛性.在本文的最后,我们给出一个数值例子,计算结果与理论分析相符合.     

四阶椭圆型方程奇异摄动问题的数值解

本文对一类四阶椭圆型方程奇异摄动问题建立了指数型拟合差分格式,并且证明了这种格式在能量范数意义下关于小参数ε的一致收敛性.最后,我们给出了数值结果.     

Multiresolution algorithms for the numerical solution of hyperbolic conservation laws

Given any scheme in conservation form and an appropriate uniform grid for the numerical solution of the initial value problem for one-dimensional hyperbolic conservation laws we describe a multiresolution algorithm that approximates this numerical solution to a prescribed tolerance in an efficient manner. To do so we consider the grid-averages of the numerical solution for a hierarchy of nested diadic grids in which the given grid is the finest, and introduce an equivalent multiresolution representation. The multiresolution representation of the numerical solution consists of its grid-averages for the coarsest grid and the set of errors in predicting the grid-averages of each level of resolution in this hierarchy from those of the next coarser one. Once the numerical solution is resolved to our satisfaction in a certain locality of some grid, then the prediction errors there are small for this particular grid and all finer ones; this enables us to compress data by setting to zero small components of the representation which fall below a prescribed tolerance. Therefore instead of computing the time-evolution of the numerical solution on the given grid we compute the time-evolution of its compressed multiresolution representation. Algorithmically this amounts to computing the numerical fluxes of the given scheme at the points of the given grid by a hierarchical algorithm which starts with the computation of these numerical fluxes at the points of the coarsest grid and then proceeds through diadic refinements to the given grid. At each step of refinement we add the values of the numerical flux at the center of the coarser cells. The information in the multiresolution representation of the numerical solution is used to determine whether the solution is locally well-resolved. When this is the case we replace the costly exact value of the numerical flux with an accurate enough approximate value which is obtained by an inexpensive interpolation from the coarser grid. The computational efficiency of this multiresolution algorithm is proportional to the rate of data compression (for a prescribed level of tolerance) that can be achieved for the numerical solution of the given scheme.     

A porous thermoviscoelastic mixture problem: numerical analysis and computational experiments

In this paper, we study, from the numerical point of view, a porous thermoviscoelastic mixture problem. The mechanical problem is written as a linear coupled system of two hyperbolic partial differential equations for the porosities and a parabolic partial differential equation for the temperature field. An existence and uniqueness result and an energy decay property are stated. Then, fully discrete approximations are introduced by using the finite element method to approximate the spatial variable and the backward Euler scheme to discretize the time derivatives. A priori error estimates are proved from which, under suitable regularity conditions, the linear convergence of the algorithm is derived. Finally, some numerical simulations are presented to demonstrate the accuracy of the approximations in an academical one-dimensional example and the behaviour of the solutions in one- and two-dimensional problems.     

MIXED FINITE ELEMENT METHOD FOR SOBOLEV EQUATIONS AND ITS ALTERNATING-DIRECTION ITERATIVE SCHEME

In this paper, we study the mixed element method for Sobolev equations. A time-discretization procedure is presented and analysed and the optimal order error estimates are derived.For convenience in practical computation, an alternating-direction iterative scheme of the mixed fi-nite element method is formulated and its stability and converbence are proved for the linear prob-lem. A numerical example is provided at the end of this paper.     

常系数扩散方程的三层九点差分格式的稳定性（英文）

In this paper,we study a numerical solution of diffusion equation.We propose a three level-nine-point implicit difference scheme and prove the difference scheme is compatible with diffusion equation,second order convergent,unconditionally stable.A numerical experiments show,the difference scheme works well inside domain,but not near the discontinuous initial-boundary points,there are still has a vibration even though it was proved unconditionally stable theoretically.We take an action to solve the disturbance,give an Algorithm,Algorithm says,we must do some primal work at the discontinuous-initial-boundary points,then starting numerical solution according the three level-nine-point implicit difference scheme we proposed in this paper.The numerical example is done once again,and there is no disturbance or vibration,our Algorithm performed well all in domain and on the boundary points with small error and good accuracy,so the Algorithm we recommended is feasible and effective.     

Numerical treatment of a mathematical model arising from a model of neuronal variability

In this paper, we describe a numerical approach based on finite difference method to solve a mathematical model arising from a model of neuronal variability. The mathematical modelling of the determination of the expected time for generation of action potentials in nerve cells by random synaptic inputs in dendrites includes a general boundary-value problem for singularly perturbed differential-difference equation with small shifts. In the numerical treatment for such type of boundary-value problems, first we use Taylor approximation to tackle the terms containing small shifts which converts it to a boundary-value problem for singularly perturbed differential equation. A rigorous analysis is carried out to obtain priori estimates on the solution of the problem and its derivatives up to third order. Then a parameter uniform difference scheme is constructed to solve the boundary-value problem so obtained. A parameter uniform error estimate for the numerical scheme so constructed is established. Though the convergence of the difference scheme is almost linear but its beauty is that it converges independently of the singular perturbation parameter, i.e., the numerical scheme converges for each value of the singular perturbation parameter (however small it may be but remains positive). Several test examples are solved to demonstrate the efficiency of the numerical scheme presented in the paper and to show the effect of the small shift on the solution behavior.     

A time-stepping scheme for inelastic collisions

We propose here a numerical scheme to compute the motion of rigid bodies with a non-elastic impact law. The method is based on a global computation of the reaction forces between bodies. Those forces, whose direction is known since we neglect friction effects, are identified at the discrete level with a scalar which plays the role of a Kuhn-Tucker multiplier associated to a first-order approximation of the non-overlapping constraint, expressed in terms of velocities. Since our original motivation is the handling of the non-overlapping constraint in fluid-particle direct simulations, we paid a special attention to stability and robustness. The scheme is proved to be stable and robust. As regards its asymptotic behaviour, a convergence result is established in the case of a single contact. Some numerical tests are presented to illustrate the properties of the algorithm. Firstly, we investigate its asymptotic behaviour in a situation of non-uniqueness, for a single particle. The two other sets of results show the good behaviour of the scheme for large time steps.     

Solving time‐periodic fractional diffusion equations via diagonalization technique and multigrid

This paper addresses numerical computation of time‐periodic diffusion equations with fractional Laplacian. Time‐periodic differential equations present fundamental challenges for numerical computation because we have to consider all the discrete solutions once in all instead of one by one. An idea based on the diagonalization technique is proposed, which yields a direct parallel‐in‐time computation for all the discrete solutions. The major computation cost is therefore reduced to solve a series of independent linear algebraic systems with complex coefficients, for which we apply a multigrid method using the damped Richardson iteration as the smoother. Such a linear solver possesses mesh‐independent convergence factor, and we make an optimization for the damping parameter to minimize such a constant convergence factor. Numerical results are provided to support our theoretical analysis.     

Analysis of Lamb waves with Spectral Finite Elements

For the analysis of the elastic wave propagation at high frequencies, the spectral finite element method (SFEM) is under investigation. The SFEM uses high-order shape functions which are composed of Lagrange polynomials with nodes at the Gauss-Lobatto quadrature points. In this way we obtain a diagonal mass matrix which makes an explicit time scheme more efficient. In a numerical example a computation by Montjoie of an elastic wave propagation within a Reissner-Mindlin (RM) model is presented. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the accumulation of eigenvalues of operator pencils connected with the problem of vibrations in a viscoelastic rod

In this paper, we study the problem of the boundary accumulation of a discrete spectrum, which is essential for a boundary-value problem of fourth order arising in the theory of small transverse vibrations in an inhomogeneous viscoelastic rod (a Kelvin—Voigt body). We establish conditions for such an accumulation and its asymptotics, which are expressed in terms of the coefficients defining the problem posed by the differential expression. The results obtained are illustrated by numerical computation data.     

A fast and robust numerical scheme for solving models of charge carrier transport and ion vacancy motion in perovskite solar cells

Drift-diffusion models that account for the motion of ion vacancies and electronic charge carriers are important tools for explaining the behaviour, and guiding the development, of metal halide perovskite solar cells. Computing numerical solutions to such models in realistic parameter regimes, where the short Debye lengths give rise to boundary layers in which the solution varies extremely rapidly, is challenging. Two suitable numerical methods, that can effectively cope with the spatial stiffness inherent to such problems, are presented and contrasted (a finite element scheme and a finite difference scheme). Both schemes are based on an appropriate choice of non-uniform spatial grid that allows the solution to be computed accurately in the boundary layers. An adaptive time step is employed in order to combat a second source of stiffness, due to the disparity in timescales between the motion of the ion vacancies and electronic charge carriers. It is found that the finite element scheme provides significantly higher accuracy, in a given compute time, than both the finite difference scheme and some previously used alternatives (Chebfun and pdepe). An example transient sweep of a current-voltage curve for realistic parameter values can be computed using this finite element scheme in only a few seconds on a standard desktop computer.     

Discrete Approximations for Singularly Perturbed Boundary Value Problems with Parabolic Layers,I

In this series of three papers we study singularly perturbed (SP) boundary value problems for equations of elliptic and parabolic type. For small values of the perturbation parameter parabolic boundary and interior layers appear in these problems. If classical discretisation methods are used, the solution of the finite difference scheme and the approximation of the diffusive flux do not converge uniformly with respect to this parameter. Using the method of special, adapted grids, we can construct difference schemes that allow approximation of the solution and the normalised diffusive flux uniformly with respect to the small parameter. We also consider singularly perturbed boundary value problems for convection-diffusion equations. Also for these problems we construct special finite difference schemes, the solution of which converges $ε$-uniformly. We study what problems appear, when classical schemes are used for the approximation of the spatial derivatives. We compare the results with those obtained by the adapted approach. Results of numerical experiments are discussed. In the three papers we first give an introduction on the general problem, and then we consider respectively (i) Problems for SP parabolic equations, for which the solution and the normalised diffusive fluxes are required; (ii) Problems for SP elliptic equations with boundary conditions of Dirichlet, Neumann and Robin type; (iii) Problems for SP parabolic equation with discontinuous boundary conditions.     

DISCRETE APPROXIMATIONS FOR SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS WITH PARABOLIC LAYERS,Ⅱ

l)ThisworkwassupportedbyNWOthroughgrantIBo7-3Go12.BOUNDAarv^LUEPRoBLEMFORELLIPTICEQUMIONwiTHMIXEDBOUNDAavCONDITION1.IntroductionInthispedwesketchavarietyofspecialmethodswhichareusedforconstructinge-unifornilyconvergelltschemes-WeshaJldemonstrateamethodwhichachieveshaprovedaccuracyforsolvingsingularlyperturbedb0undaryvalueproblemforeiliPicequatiouswithparabolicboundarylayers-InSecti0n4weshallintroduceanaturalclass,B,oftritefferenceschemes,inwhich(bytheabovementi0nedaP…     

Discrete Approximations for Singularly Perturbed Boundary Value Problems with Parabolic Layers,III

1.IntroductionThesolution0fpartialdifferentiaJequationsthataresingularlyperturbedand/orhavediscontinu0usboundaryconditionsgenerallyhave0nlylimitedsmoothness.DuetothisfaCtdndcultiesaPpearwhenwesolvethesepr0blemsbynumericalmethods.Forexampleforregularparab0licequationswithdiscontinuousboundaryconditions,classicalmethods(FDMorFEM)onregularrectangulargridsd0n0tconvergeintheIoo-normonadomainthatincludesaneighbourhood0fthediscontinulty[8,9,4].Iftheparametermultiplyingthehighest-orderderivativeva…     

New general guidance method in constrained optimal control,part 1: Numerical method

A very fast numerical method is developed for the computation of neighboring optimum feedback controls. This method is applicable to a general class of optimal control problems (for example, problems including inequality constraints and discontinuities) and needs no on-line computation, except for one matrix-vector multiplication. The method is based on the so-called accessory minimum problem. The necessary conditions for this auxiliary optimal control problem form a linear multipoint boundary-value problem with linear jump conditions, which is especially well suited for numerical treatment. In the second part of this paper, the performance of the guidance scheme is shown for the heating-constrained cross-range maximization problem of a space-shuttle-orbiter-type vehicle.This research was supported in part by the Deutsche Forschungsgemeinschaft under the Schwerpunktprogramm Anwendungsbezogene Optimierung und Steuerung.The authors wish to express their sincere and grateful appreciation to Professor Roland Bulirsch who encouraged this work.     

A fourth‐order compact finite difference scheme for solving an N‐carrier system with Neumann boundary conditions

In this study, we develop a fourth‐order compact finite difference scheme for solving a model of energy exchanges in a generalized N‐carrier system with heat sources and Neumann boundary conditions, which extends the concept of the well‐known parabolic two‐step model for microheat transfer. By using the matrix analysis, the compact finite difference numerical scheme is shown to be unconditionally stable. The accuracy of the solution obtained by the scheme is tested by a numerical example. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Computation of Current-Voltage Characteristics in a Semiconductor Device using Arc-length Continuation

This paper is concerned with the computation of semiconductordevice current—voltage characteristics. We describe analgorithm which allows the computation of characteristics bycontinuation in a parameter which approximates the arc lengthof the characteristic. The use of this parameterization allowsthe characteristic to continue beyond snap-back voltages, whilecontinuation in the voltage fails past snap-back voltages. Wediscuss the implementation of the parameterization and givea numerical example.