On efficient numerical methods for an initial-boundary value problem with nonlocal boundary conditions

Many physical phenomena are modeled by nonclassical hyperbolic boundary value problems with nonlocal boundary conditions. In this paper, the problem of solving the one-dimensional wave equation subject to given initial and non-local boundary conditions is considered. These non-local conditions arise mainly when the data on the boundary cannot be measured directly. Several finite difference methods with low order have been proposed in other papers for the numerical solution of this one dimensional non-classic boundary value problem. Here, we derive a new family of efficient three-level algorithms with higher order to solve the wave equation and also use a Simpson formula with higher order to approximate the integral conditions. Additionally, the fourth-order formula is also adapted to nonlinear equations, in particular to the well-known nonlinear Klein–Gordon equations which many physical problems can be modeled with. Numerical results are presented and are compared with some existing methods showing the efficiency of the new algorithms.     

A parametrization method for solving nonlinear two-point boundary value problems

A sharper version of the local Hadamard theorem on the solvability of nonlinear equations is proved. Additional parameters are introduced, and a two-parameter family of algorithms for solving nonlinear two-point boundary value problems is proposed. Conditions for the convergence of these algorithms are given in terms of the initial data. Using the right-hand side of the system of differential equations and the boundary conditions, equations are constructed from which initial approximations to the unknown parameters can be found. A criterion is established for the existence of an isolated solution to a nonlinear two-point boundary value problem. This solution is shown to be a continuous function of the data specifying the problem.     

Two‐grid variational multiscale algorithms for the stationary incompressible Navier‐Stokes equations with friction boundary conditions

Two‐grid variational multiscale (VMS) algorithms for the incompressible Navier‐Stokes equations with friction boundary conditions are presented in this article. First, one‐grid VMS algorithm is used to solve this problem and some error estimates are derived. Then, two‐grid VMS algorithms are proposed and analyzed. The algorithms consist of nonlinear problem on coarse grid and linearized problem (Stokes problem or Oseen problem) on fine grid. Moreover, the stability and convergence of the present algorithms are established. Finally, Numerical results are shown to confirm the theoretical analysis. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 546–569, 2017     

Boundary control problems for quasilinear systems of hyperbolic equations

For quasilinear systems of hyperbolic equations, the nonclassical boundary value problem of controlling solutions with the help of boundary conditions is considered. Previously, this problem was extensively studied in the case of the simplest hyperbolic equations, namely, the scalar wave equation and certain linear systems. The corresponding problem formulations and numerical solution algorithms are extended to nonlinear (quasilinear and conservative) systems of hyperbolic equations. Some numerical (grid-characteristic) methods are considered that were previously used to solve the above problems. They include explicit and implicit conservative difference schemes on compact stencils that are linearizations of Godunov’s method. The numerical algorithms and methods are tested as applied to well-known linear examples.     

Numerical algorithms for diffusion-reaction problems with non-classical conditions

Parabolic equations with nonlocal boundary conditions have been given considerable attention in recent years. In this paper new high-order algorithms for the linear diffusion-reaction problem are derived. The convergence of the new schemes is studied and numerical examples are given to show the efficiency of the new methods to solve linear and nonlinear diffusion-reaction equations with these non classical conditions.     

Direct and inverse boundary value problems for models of stationary reaction–convection–diffusion

Direct and inverse boundary value problems for models of stationary reaction–convection–diffusion are investigated. The direct problem consists in finding a solution of the corresponding boundary value problem for given data on the boundary of the domain of the independent variable. The peculiarity of the direct problem consists in the inhomogeneity and irregularity of mixed boundary data. Solvability and stability conditions are specified for the direct problem. The inverse boundary value problem consists in finding some traces of the solution of the corresponding boundary value problem for given standard and additional data on a certain part of the boundary of the domain of the independent variable. The peculiarity of the inverse problem consists in its ill-posedness. Regularizing methods and solution algorithms are developed for the inverse problem.     

Constructive methods for obtaining the solution of the periodic boundary value problem for a system of matrix differential equations of Riccati type

In the nonsingular case, we obtain sufficient coefficient conditions for the unique solvability of the periodic boundary value problem for a system of matrix differential equations of Riccati type. We develop efficient algorithms for constructing the solution.     

Wavelet-Galerkin method for solving parabolic equations in finite domains

A novel wavelet-Galerkin method tailored to solve parabolic equations in finite domains is presented. The emphasis of the paper is on the development of the discretization formulations that are specific to finite domain parabolic equations with arbitrary boundary conditions based on weak form functionals. The proposed method also deals with the development of algorithms for computing the associated connection coefficients at arbitrary points. Here the Lagrange multiplier method is used to enforce the essential boundary conditions. The numerical results on a two-dimensional transient heat conducting problem are used to validate the proposed wavelet-Galerkin algorithm as an effective numerical method to solve finite domain parabolic equations.     

Low cost a posteriori error estimators for an augmented mixed FEM in linear elasticity

We consider an augmented mixed finite element method applied to the linear elasticity problem and derive a posteriori error estimators that are simpler and easier to implement than the ones available in the literature. In the case of homogeneous Dirichlet boundary conditions, the new a posteriori error estimator is reliable and locally efficient, whereas for non-homogeneous Dirichlet boundary conditions, we derive an a posteriori error estimator that is reliable and satisfies a quasi-efficiency bound. Numerical experiments illustrate the performance of the corresponding adaptive algorithms and support the theoretical results.     

A method for solving an exterior three-dimensional boundary value problem for the Laplace equation

We develop and experimentally study the algorithms for solving three-dimensionalmixed boundary value problems for the Laplace equation in unbounded domains. These algorithms are based on the combined use of the finite elementmethod and an integral representation of the solution in a homogeneous space. The proposed approach consists in the use of the Schwarz alternating method with consecutive solution of the interior and exterior boundary value problems in the intersecting subdomains on whose adjoining boundaries the iterated interface conditions are imposed. The convergence of the iterative method is proved. The convergence rate of the iterative process is studied analytically in the case when the subdomains are spherical layers with the known exact representations of all consecutive approximations. In this model case, the influence of the algorithm parameters on the method efficiency is analyzed. The approach under study is implemented for solving a problem with a sophisticated configuration of boundaries while using a high precision finite elementmethod to solve the interior boundary value problems. The convergence rate of the iterations and the achieved accuracy of the computations are illustrated with some numerical experiments.     

Constrained order packing: comparison of heuristic approaches for a new bin packing problem

Bin packing problems consist of allocating a set of small parts to a set of large bins by minimizing the number of used bins. Although several boundary conditions have been stated in the past, for example conflicts or restrictions on cutting and rotations, we introduce a set of constraints, which lead to a new problem structure. These constraints are motivated by the precast-concrete-part industry and represent requirements on the ordering of parts and their positions, machinery restrictions and due dates. Furthermore, we solve the problem using several heuristic approaches that are based on algorithms for the standard bin packing problem. Therefore, existing concepts are classified and adapted to fit the new problem, including Simulated Annealing, Genetic Algorithms, Tabu Search methods and SubSetSum based search routines. Finally, all proposed algorithms are tested and obtained results are discussed.     

An iterative penalty method for the least squares solution of boundary value problems

This article is concerned with iterative techniques for linear systems of equations arising from a least squares formulation of boundary value problems. In its classical form, the solution of the least squares method is obtained by solving the traditional normal equation. However, for nonsmooth boundary conditions or in the case of refinement at a selected set of interior points, the matrix associated with the normal equation tends to be ill-conditioned. In this case, the least squares method may be formulated as a Powell multiplier method and the equations solved iteratively. Therein we use and compare two different iterative algorithms. The first algorithm is the preconditioned conjugate gradient method applied to the normal equation, while the second is a new algorithm based on the Powell method and formulated on the stabilized dual problem. The two algorithms are first compared on a one-dimensional problem with poorly conditioned matrices. Results show that, for such problems, the new algorithm gives more accurate results. The new algorithm is then applied to a two-dimensional steady state diffusion problem and a boundary layer problem. A comparison between the least squares method of Bramble and Schatz and the new algorithm demonstrates the ability of the new method to give highly accurate results on the boundary, or at a set of given interior collocation points without the deterioration of the condition number of the matrix. Conditions for convergence of the proposed algorithm are discussed. © 1997 John Wiley & Sons, Inc.     

A parareal approach of semi‐linear parabolic equations based on general waveform relaxation

We present a parareal approach of semi‐linear parabolic equations based on general waveform relaxation (WR) at the partial differential equation (PDE) level. An algorithm for initial‐boundary value problem and two algorithms for time‐periodic boundary value problem are constructed. The convergence analysis of three algorithms are provided. The results show that the algorithm for initial‐boundary value problem is superlinearly convergent while both algorithms for the time‐periodic boundary value problem linearly converge to the exact solutions at most. Numerical experiments show that the parareal algorithms based on general WR at the PDE level, compared with the parareal algorithm based on the classical WR at the ordinary differential equations (ODEs) level (the PDEs is discretized into ODEs), require much fewer number of iterations to converge.     

A new type of boundary treatment for wave propagation

Numerical approximation of wave propagation can be done very efficiently on uniform grids. The Yee scheme is a good example. A serious problem with uniform grids is the approximation of boundary conditions at a boundary not aligned with the grid. In this paper, boundary conditions are introduced by modifying appropriate material coefficients at a few grid points close to the embedded boundary. This procedure is applied to the Yee scheme and the resulting method is proven to be \(L^2\)-stable in one space dimension. Depending on the boundary approximation technique it is of first or second order accuracy even if the boundary is located at an arbitrary point relative to the grid. This boundary treatment is applied also to a higher order discretization resulting in a third order accurate method. All algorithms have the same staggered grid structure in the interior as well as across the boundaries for efficiency. A numerical example with the extension to two space dimensions is included.     

Supplementary optimality properties of gradient-restoration algorithms for optimal control problems

In this paper, sequential gradient-restoration algorithms for optimal control problems are considered, and attention is focused on the gradient phase. It is shown that the Lagrange multipliers associated with the gradient phase not only solve the auxiliary minimization problem of the gradient phase, but are also endowed with a supplementary optimality property: they minimize the error in the optimality conditions, subject to the multiplier differential equations and boundary conditions, for given state, control, and parameter.Dedicated to R. BellmanThis work was supported by the National Science Foundation, Grant No. ENG-79-18667.     

Finite difference reaction-diffusion systems with coupled boundary conditions and time delays

This paper is concerned with finite difference solutions of a coupled system of reaction-diffusion equations with nonlinear boundary conditions and time delays. The system is coupled through the reaction functions as well as the boundary conditions, and the time delays may appear in both the reaction functions and the boundary functions. The reaction-diffusion system is discretized by the finite difference method, and the investigation is devoted to the finite difference equations for both the time-dependent problem and its corresponding steady-state problem. This investigation includes the existence and uniqueness of a finite difference solution for nonquasimonotone functions, monotone convergence of the time-dependent solution to a maximal or a minimal steady-state solution for quasimonotone functions, and local and global attractors of the time-dependent system, including the convergence of the time-dependent solution to a unique steady-state solution. Also discussed are some computational algorithms for numerical solutions of the steady-state problem when the reaction function and the boundary function are quasimonotone. All the results for the coupled reaction-diffusion equations are directly applicable to systems of parabolic-ordinary equations and to reaction-diffusion systems without time delays.     

Numerical Solution of Boundary Value Problems for Selfadjoint Differential Equations of 2nth Order

The paper is devoted to solving boundary value problems for self-adjoint linear differential equations of 2nth order in the case that the corresponding differential operator is self-adjoint and positive semidefinite. The method proposed consists in transforming the original problem to solving several initial value problems for certain systems of first order ODEs. Even if this approach may be used for quite general linear boundary value problems, the new algorithms described here exploit the special properties of the boundary value problems treated in the paper. As a consequence, we obtain algorithms that are much more effective than similar ones used in the general case. Moreover, it is shown that the algorithms studied here are numerically stable.     

Numerical methods for fourth‐order nonlinear elliptic boundary value problems

The aim of this article is to present several computational algorithms for numerical solutions of a nonlinear finite difference system that represents a finite difference approximation of a class of fourth‐order elliptic boundary value problems. The numerical algorithms are based on the method of upper and lower solutions and its associated monotone iterations. Three linear monotone iterative schemes are given, and each iterative scheme yields two sequences, which converge monotonically from above and below, respectively, to a maximal solution and a minimal solution of the finite difference system. This monotone convergence property leads to upper and lower bounds of the solution in each iteration as well as an existence‐comparison theorem for the finite difference system. Sufficient conditions for the uniqueness of the solution and some techniques for the construction of upper and lower solutions are obtained, and numerical results for a two‐point boundary‐value problem with known analytical solution are given. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:347–368, 2001     

Extrapolation algorithms for solving nonlinear boundary integral equations by mechanical quadrature methods

We study the numerical solution procedure for two-dimensional Laplace’s equation subjecting to non-linear boundary conditions. Based on the potential theory, the problem can be converted into a nonlinear boundary integral equations. Mechanical quadrature methods are presented for solving the equations, which possess high accuracy order O(h ³) and low computing complexities. Moreover, the algorithms of the mechanical quadrature methods are simple without any integration computation. Harnessing the asymptotical compact theory and Stepleman theorem, an asymptotic expansion of the errors with odd powers is shown. Based on the asymptotic expansion, the h ³ −Richardson extrapolation algorithms are used and the accuracy order is improved to O(h ⁵). The efficiency of the algorithms is illustrated by numerical examples.     

Stable second-order finite-difference methods for linear initial-boundary-value problems

Finite-difference methods of second order at the boundary points are presented for problems with one-dimensional second-order hyperbolic and parabolic equations with mixed boundary conditions. These methods do not require information at points outside the region of consideration. The linear stability of the algorithms developed is investigated. Numerical experiments are given for illustrating the accuracy and stability of the methods. Though the focus is on homogeneous boundary conditions, finite-difference methods with non-homogeneous mixed boundary conditions are also developed. To show the potential of the methods developed, in terms of CPU time, a comparison is made with the Crank–Nicolson method.