The use of compact boundary value method for the solution of two-dimensional Schrödinger equation

In this paper, a high-order and accurate method is proposed for solving the unsteady two-dimensional Schrödinger equation. We apply a compact finite difference approximation of fourth-order for discretizing spatial derivatives and a boundary value method of fourth-order for the time integration of the resulting linear system of ordinary differential equations. The proposed method has fourth-order accuracy in both space and time variables. Moreover this method is unconditionally stable due to the favorable stability property of boundary value methods. The results of numerical experiments are compared with analytical solutions and with those provided by other methods in the literature. These results show that the combination of a compact finite difference approximation of fourth-order and a fourth-order boundary value method gives an efficient algorithm for solving the two dimensional Schrödinger equation.     

Numerical computation of neighboring optimum feedback control schemes in real-time

A modification of the theory of neighboring extremals is presented which leads to a new formulation of a linear boundary value problem for the perturbation of the state and adjoint variables around a reference trajectory. On the basis of the multiple shooting algorithm, a numerical method for stable and efficient computation of perturbation feedback schemes is developed. This method is then applied to guidance problems in astronautics. Using as much stored a priori information about the precalculated flight path as possible, the only computational work to be done on the board computer for the computation of a regenerated optimal control program is a single integration of the state differential equations and the solution of a few small systems of linear equations. The amount of computation is small enough to be carried through on modern board computers for real-time. Nevertheless, the controllability region is large enough to compensate realistic flight disturbances, so that optimality is preserved.     

The solution and sensitivity of a general optimal control problem

A solution method for the general optimal control problem is presented. This can be used to solve optimal control problems for which the system dynamics are not necessarily described by differential or difference equations. Having obtained the solution it is of theoretical and practical interest to investigate the sensitivity of the optimal trajectory to perturbations. Indicators of this sensitivity are the adjoint variables derived in the maximum principle. A method of deriving the adjoint variables from the solution is described. To illustrate the solution method and the determination of the adjoint variables a problem in the urban housing market is used.     

Order h4 difference methods for a class of singular two space elliptic boundary value problems

In this article, we derive difference methods of O(h⁴) for solving the system of two space nonlinear elliptic partial differential equations with variable coefficients having mixed derivatives on a uniform square grid using nine grid points. We obtain two sets of fouth-order difference methods; one in the absence of mixed derivatives, second when the coefficients of u_xy are not equal to zero and the coefficients of u_xx and u_yy are equal. There do not exist fourth-order schemes involving nine grid points for the general case. The method having two variables has been tested on two-dimensional viscous, incompressible steady-state Navier-Stokes' model equations in polar coordinates. The proposed difference method for scalar equation is also applied to the Poisson's equation in polar coordinates. Some numerical examples are provided to illustrate the fourth-order convergence of the proposed methods.     

An expanded mixed covolume method for elliptic problems

We consider the mixed covolume method combining with the expanded mixed element for a system of first‐order partial differential equations resulting from the mixed formulation of a general self‐adjoint elliptic problem with a full diffusion tensor. The system can be used to model the transport of a contaminant carried by a flow in porous media. We use the lowest order Raviart‐Thomas mixed element space. We show the first‐order error estimate for the approximate solution in L² norm. We show the superconvergence both for pressure and velocity in certain discrete norms. We also get a finite difference scheme by using proper approximate integration formulas. Finally we give some numerical examples. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Error estimates of triangular mixed finite element methods for quasilinear optimal control problems

The goal of this paper is to study a mixed finite element approximation of the general convex optimal control problems governed by quasilinear elliptic partial differential equations. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a priori error estimates both for the state variables and the control variable. Finally, some numerical examples are given to demonstrate the theoretical results.     

Optimal Control of One-Dimensional Partial Differential Algebraic Equations with Applications

We present an approach to compute optimal control functions in dynamic models based on one-dimensional partial differential algebraic equations (PDAE). By using the method of lines, the PDAE is transformed into a large system of usually stiff ordinary differential algebraic equations and integrated by standard methods. The resulting nonlinear programming problem is solved by the sequential quadratic programming code NLPQL. Optimal control functions are approximated by piecewise constant, piecewise linear or bang-bang functions. Three different types of cost functions can be formulated. The underlying model structure is quite flexible. We allow break points for model changes, disjoint integration areas with respect to spatial variable, arbitrary boundary and transition conditions, coupled ordinary and algebraic differential equations, algebraic equations in time and space variables, and dynamic constraints for control and state variables. The PDAE is discretized by difference formulae, polynomial approximations with arbitrary degrees, and by special update formulae in case of hyperbolic equations. Two application problems are outlined in detail. We present a model for optimal control of transdermal diffusion of drugs, where the diffusion speed is controlled by an electric field, and a model for the optimal control of the input feed of an acetylene reactor given in form of a distributed parameter system.     

Fourth-order compact solution of the nonlinear Klein-Gordon equation

In this work we propose a fourth-order compact method for solving the one-dimensional nonlinear Klein-Gordon equation. We apply a compact finite difference approximation of fourth-order for discretizing spatial derivative and a fourth-order A-stable diagonally-implicit Runge-Kutta-Nyström (DIRKN) method for the time integration of the resulting nonlinear second-order system of ordinary differential equations. The proposed method has fourth order accuracy in both space and time variables and is unconditionally stable. Numerical results obtained from solving several problems possessing periodic, kinks, single and double-soliton waves show that the combination of a compact finite difference approximation of fourth order and a fourth-order A-stable DIRKN method gives an efficient algorithm for solving these problems.     

Greedy sparse linear approximations of functionals from nodal data

In many numerical algorithms, integrals or derivatives of functions have to be approximated by linear combinations of function values at nodes. This ranges from numerical integration to meshless methods for solving partial differential equations. The approximations should use as few nodal values as possible and at the same time have a smallest possible error. For each fixed set of nodes and each fixed Hilbert space of functions with continuous point evaluation, e.g. a fixed Sobolev space, there is an error–optimal method available using the reproducing kernel of the space. But the choice of the nodes is usually left open. This paper shows how to select good nodes adaptively by a computationally cheap greedy method, keeping the error optimal in the above sense for each incremental step of the node selection. This is applied to interpolation, numerical integration, and numerical differentiation. The latter case is particularly important for the design of meshless methods with sparse generalized stiffness matrices. The greedy algorithm is described in detail, and numerical examples are provided. In contrast to the usual practice, the greedy method does not always use nearest neighbors for local approximations of function values and derivatives. Furthermore, it avoids multiple points from clusters and it is better conditioned than choosing nearest neighbors.     

Adaptive finite element methods for mixed control-state constrained optimal control problems for elliptic boundary value problems

Mixed control-state constraints are used as a relaxation of originally state constrained optimal control problems for partial differential equations to avoid the intrinsic difficulties arising from measure-valued multipliers in the case of pure state constraints. In particular, numerical solution techniques known from the pure control constrained case such as active set strategies and interior-point methods can be used in an appropriately modified way. However, the residual-type a posteriori error estimators developed for the pure control constrained case can not be applied directly. It is the essence of this paper to show that instead one has to resort to that type of estimators known from the pure state constrained case. Up to data oscillations and consistency error terms, they provide efficient and reliable estimates for the discretization errors in the state, a regularized adjoint state, and the control. A documentation of numerical results is given to illustrate the performance of the estimators.     

Necessary conditions without differentiability assumptions in unilateral control problems

We derive two theorems combining existence with necessary conditions for the relaxed unilateral problem of the optimal control of ordinary differential equations in which the functions that define the problem are Lipschitz-continuous in the state variables. These theorems generalize the results presented in a previous paper [8] by the addition of unilateral constraints on the state and control functions. As in that paper, the new necessary conditions have a canonical form obtained by replacing, in the “customary” conditions, the partial derivatives with respect to the state variables by finite difference quotients at neighboring arguments, and then applying limiting processes and convexification. More general necessary conditions are also obtained in terms of the representations of the Lipschitz-continuous functions as compositions.     

Analysis of expanded mixed methods for fourth-order elliptic problems

The recently proposed expanded mixed formulation for numerical solution of second-order elliptic problems is here extended to fourth-order elliptic problems. This expanded formulation for the differential problems under consideration differs from the classical formulation in that three variables are treated, i.e., the displacement, the stress, and the moment tensors. It works for the case where the coefficient of the differential equations is small and does not need to be inverted, or for the case in which the stress tensor of the equations does not need to be symmetric. Based on this new formulation, various mixed finite elements for fourth-order problems are considered; error estimates of quasi-optimal or optimal order depending upon the mixed elements are derived. Implementation techniques for solving the linear system arising from these expanded mixed methods are discussed, and numerical results are presented. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 483–503, 1997     

Implicit difference methods for evolution functional differential equations

A general theory of implicit difference schemes for nonlinear functional differential equations with initial boundary conditions is presented. A theorem on error estimates of approximate solutions for implicit functional difference equations of the Volterra type with an unknown function of several variables is given. This general result is employed to investigate the stability of implicit difference schemes generated by first-order partial differential functional equations and by parabolic problems. A comparison technique with nonlinear estimates of the Perron type for given functions with respect to the functional variable is used.     

A direct approach to the finite element solution of elliptic optimal control problems

A general framework is developed for the finite element solution of optimal control problems governed by elliptic nonlinear partial differential equations. Typical applications are steady‐state problems in nonlinear continuum mechanics, where a certain property of the solution (a function of displacements, temperatures, etc.) is to be minimized by applying control loads. In contrast to existing formulations, which are based on the “adjoint state,” the present formulation is a direct one, which does not use adjoint variables. The formulation is presented first in a general nonlinear setting, then specialized to a case leading to a sequence of quadratic programming problems, and then specialized further to the unconstrained case. Linear governing partial differential equations are also considered as a special case in each of these categories. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15:371–388, 1999     

The geometry of a quasilinear system of two partial differential equations containing the first and second partial derivatives of two functions in two independent variables

The geometry of a system of two partial differential equations containing the first and second partial derivatives of two functions in two independent variables is studied by using the Cartan method of invariant forms and the group-theoretic method of extensions and enclosings due to G. F. Laptev (for finite groups) and A. M. Vasil’ev (for infinite groups). Systems of quasilinear equations with the first and second partial derivatives of two functions u and v in two independent variables x and y are classified.     

A family of ELLAM schemes for advection-diffusion-reaction equations and their convergence analyses

A family of ELLAM (Eulerian–Lagrangian localized adjoint method) schemes is developed and analyzed for linear advection-diffusion-reaction transport partial differential equations with any combination of inflow and outflow Dirichlet, Neumann, or flux boundary conditions. The formulation uses space-time finite elements, with edges oriented along Lagrangian flow paths, in a time–stepping procedure, where space-time test functions are chosen to satisfy a local adjoint condition. This allows Eulerian–Lagrangian concepts to be applied in a systematic mass-conservative manner, yielding numerical schemes defined at each discrete time level. Optimal-order error estimates and superconvergence results are derived. Numerical experiments are performed to verify the theoretical estimates. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 739–780, 1998     

Approximation of weak adjoints by reverse automatic differentiation of BDF methods

We shed light on the relation between the discrete adjoints of multistep backward differentiation formula (BDF) methods and the solution of the adjoint differential equation. To this end, we develop a functional-analytic framework based on a constrained variational problem and introduce the notion of weak adjoint solutions of ordinary differential equations. We devise a Petrov-Galerkin finite element (FE) interpretation of the BDF method and its discrete adjoint scheme obtained by reverse internal numerical differentiation. We show how the FE approximation of the weak adjoint is computed by the discrete adjoint scheme and prove its convergence in the space of normalized functions of bounded variation. We also show convergence of the discrete adjoints to the classical adjoints on the inner time interval. Finally, we give numerical results for non-adaptive and fully adaptive BDF schemes. The presented framework opens the way to carry over techniques on global error estimation from FE methods to BDF methods.     

A variable gravity model of the otolith membrane

This paper describes a finite difference model of the otolith membrane which allows the acceleration due to gravity to vary, thus simulating conditions of gravity on the lunar and planetary surfaces.The differential coefficients of the second-order system of elliptic partial differential equations governing the steady state displacements of points of the membrane are replaced by finite differences. The resulting system of difference equations is seen to be consistent with the system of differential equations and to have a truncation error of order four.A close approximation to the physical boundary of a typical otolith membrane is used and two sets of numerical experiments are carried out which which simulate rotations of the membrane on the Moon and on a number of planets.The displacements at thirty nodes of the membrane are computed by solving the linear system of sixty equations obtained by applying the difference equations to each of the thirty nodes. The numerical results obtained are seen to be in general agreement with experimental results reported in the literature.     

Ostrowski’s fourth-order iterative method speedily solves cubic equations of state

Pressure-volume-temperature (P-V-T) data are required in simulating chemical plants because the latter usually involve production, separation, transportation, and storage of fluids. In the absence of actual experimental data, the pertinent mathematical model must rely on phase behaviour prediction by the so-called equations of state (EOS). When the plant model is a combination of differential and algebraic equations, simulation generally relies on numerical integration which proceeds in a piecewise fashion unless an approximate solution is needed at a single point. Needless to say, the constituent algebraic equations must be efficiently re-solved before each update of derivatives. Now, Ostrowski’s fourth-order iterative technique is a partial substitution variant of Newton’s popular second-order method. Although simple and powerful, this two-point variant has been utilised very little since its publication over forty years ago. After a brief introduction to cubic equations of state and their solution, this paper solves five of them. The results clearly demonstrate the superiority of Ostrowski’s method over Newton’s, Halley’s, and Chebyshev’s solvers.     

State theory of linear hereditary differential systems

In this paper we present a state theory for a class of linear functional differential equations of the retarded type considered by Delfour and Mitter (J. Differential Equations, 18 1975, 18–28) with initial functions in the product space M^p = X × L^p(?b, 0; X). Roughly speaking, the state at time t is a piece of trajectory defined over an interval [t ? b, t] for a fixed b > 0. From a study of the properties of the state in M^p an operational differential equation, the so-called state equation, is derived in order to describe its evolution. An adjoint state equation is also introduced for the adjoint state and the connection between solutions of the hereditary adjoint system and those of the adjoint state equation is established. All this provides the appropriate framework for the solution and the numerical approximation of the associated linear-quadratic optimal control and filtering problems.