Solution of the Navier-Stokes equations by the spectral-difference method

The Jacobi polynomial collocation method is extended to two and three dimensions by superimposing the individual one-dimensional expansions. Two innovative ideas are introduced in this article. The first is the cornerless/edgeless computational grid, and the second is the modified compressibility method, which is an iterative pressure solver. To evaluate the new method's applicability in solving the Navier-Stokes equation, the lid-driven cavity flow problem was solved. Two configurations were considered, the square cavity and the rectangular cavity with an aspect ratio of 2. A comparison of the center-line velocity profiles from two- and three-dimensional simulations at a Reynolds number of 1000 is provided for each of the configurations. The center-line velocity comparisons showed good quantitative agreement with previous studies. © 1994 John Wiley & Sons, Inc.     

Least-squares spectral collocation with the overlapping Schwarz method for the incompressible Navier–Stokes equations

A least-squares spectral collocation scheme is combined with the overlapping Schwarz method. The methods are succesfully applied to the incompressible Navier–Stokes equations. The collocation conditions and the interface conditions lead to an overdetermined system which can be efficiently solved by least-squares. The solution technique will only involve symmetric positive definite linear systems. The overlapping Schwarz method is used for the iterative solution. For parallel implementation the subproblems are solved in a checkerboard manner. Our approach is successfully applied to the lid-driven cavity flow problem. Only a few Schwarz iterations are necessary in each time step. Numerical simulations confirm the high accuracy of our spectral least-squares scheme.     

On the numerical treatment of the eigenparameter dependent boundary conditions

In this paper, we consider the numerical treatment of singular eigenvalue problems supplied with eigenparameter dependent boundary conditions using spectral methods. On the one hand, such boundary conditions hinder the construction of test and trial space functions which could incorporate them and thus providing well-conditioned Galerkin discretization matrices. On the other hand, they can generate surprising behavior of the eigenvectors hardly detected by analytic methods. These singular problems are often indirectly approximated by regular ones. We argue that spectral collocation as well as tau method offer remedies for the first two issues and provide direct and efficient treatment to such problems. On a finite domain, we consider the so-called Petterson-König’s rod eigenvalue problem and on the half line, we take into account the Charney’s baroclinic stability problem and the Fourier eigenvalue problem. One boundary condition in these problems depends on the eigenparameter and additionally, this also could depend on some physical parameters. The Chebyshev collocation based on both, square and rectangular differentiation and a Chebyshev tau method are used to discretize the first problem. All these schemes cast the problems into singular algebraic generalized eigenvalue ones which are solved by the QZ and/or Arnoldi algorithms as well as by some target oriented Jacobi-Davidson methods. Thus, the spurious eigenvalues are completely eliminated. The accuracy of square Chebyshev collocation is roughly estimated and its order of approximation with respect to the eigenvalue of interest is determined. For the problems defined on the half line, we make use of the Laguerre-Gauss-Radau collocation. The method proved to be reliable, accurate, and stable with respect to the order of approximation and the scaling parameter.     

Numerical solution of partial differential equations on curved domains by collocation

The application of the finite element collocation method to two- and three-dimensional partial differential equations is hampered by the fact that its accuracy depends largely on the position of the collocation points. The method has, in the past, thus mainly been applied to two-dimensional differential equations defined on rectangular domains. In this case the method yields an optimal global convergence rate, when the Gauss-Legendre quadrature points are used as collocation points. It is shown in this paper that the same accuracy can also be achieved in the case of differential equations defined on nonrectangular domains. The only prerequisite for this is to solve the differential equation on a rectangular domain, mapped onto the nonrectangular domain of the differential equation by a bilinear blended map. © 1993 John Wiley & Sons, Inc.     

A quadrature finite element Galerkin scheme for a biharmonic problem on a rectangular polygon

A quadrature Galerkin scheme with the Bogner–Fox–Schmit element for a biharmonic problem on a rectangular polygon is analyzed for existence, uniqueness, and convergence of the discrete solution. It is known that a product Gaussian quadrature with at least three‐points is required to guarantee optimal order convergence in Sobolev norms. In this article, optimal order error estimates are proved for a scheme based on the product two‐point Gaussian quadrature by establishing a relation with an underdetermined orthogonal spline collocation scheme. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Numerical implementation of the spline-collocation method for bending of rectangular plates

Different approaches are considered to the application of the spline-collocation method to bending of rectangular plates. The effect of the choice of collocation nodes on the order of accuracy of the method is investigated. Numerical results obtained with different arrangements of collocation points are reported.Institute of Mechanics, Ukrainian Academy of Sciences. Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 68, pp. 93–100, 1989.     

A collocation method based on the Bessel functions of the first kind for singular perturbated differential equations and residual correction

In this paper, a collocation method is given to solve singularly perturbated two‐point boundary value problems. By using the collocation points, matrix operations and the matrix relations of the Bessel functions of the first kind and their derivatives, the boundary value problem is converted to a system of the matrix equations. By solving this system, the approximate solution is obtained. Also, an error problem is constructed by the residual function, and it is solved by the presented method. Thus, the error function is estimated, and the approximate solutions are improved. Finally, numerical examples are given to show the applicability of the method, and also, our results are compared by existing results. Copyright © 2014 JohnWiley & Sons, Ltd.     

A decomposition approach for optimal control problems with integer innerpoint state constraints

A large class of optimal control problems for hybrid dynamic systems can be formulated as mixed‐integer optimal control problems (MIOCPs). A decomposition approach is suggested to solve a special subclass of MIOCPs with mixed integer inner point state constraints. It is the intrinsic combinatorial complexity of the discrete variables in addition to the high nonlinearity of the continuous optimal control problem that forms the challenges in the theoretical and numerical solution of MIOCPs. During the solution procedure the problem is decomposed at the inner time points into a multiphase problem with mixed integer boundary constraints and phase transitions at unknown switching points. Due to a discretization of the state space at the switching points the problem can be decoupled into a family of continuous optimal control problems (OCPs) and a problem similar to the asymmetric group traveling salesman problem (AGTSP). The OCPs are transcribed by direct collocation to large‐scale nonlinear programming problems, which are solved efficiently by an advanced SQP method. The results are used as weights for the edges of the graph of the corresponding TSP‐like problem, which is solved by a Branch‐and‐Cut‐and‐Price (BCP) algorithm. The proposed approach is applied to a hybrid optimal control benchmark problem for a motorized traveling salesman. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonconforming spline collocation methods in irregular domains

This article studies a class of nonconforming spline collocation methods for solving elliptic PDEs in an irregular region with either triangular or quadrilateral partition. In the methods, classical Gaussian points are used as matching points and the special quadrature points in a triangle or quadrilateral element are used as collocation points. The solution and its normal derivative are imposed to be continuous at the marching points. The authors present theoretically the existence and uniqueness of the numerical solution as well as the optimal error estimate in H¹‐norm for a spline collocation method with rectangular elements. Numerical results confirm the theoretical analysis and illustrate the high‐order accuracy and some superconvergence features of methods. Finally the authors apply the methods for solving two physical problems in compressible flow and linear elasticity, respectively. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

On the numerical method for solving a hypersingular integral equation with the computation of the solution gradient

We consider a linear integral equation, which arises when solving the Neumann boundary value problem for the Laplace equation with the representation of the solution in the form of a double layer potential, with a hypersingular integral treated in the sense of Hadamard finite value. We consider the case in which the exterior or interior problem is solved in a domain whose boundary is a closed smooth surface and the integral equation is written over that surface. A numerical scheme for solving the integral equation is constructed with the use of quadrature formulas of the type of the method of discrete singularities with a regularization for the use of an irregular grid. We prove the convergence, uniform over the grid points, of the numerical solutions to the exact solution of the hypersingular equation and, in addition, the uniform convergence of the values of the approximate finite-difference derivative operator on the numerical solution to the values on the projection of the exact solution onto the subspace of grid functions with nodes at the collocation points.     

On the determination of boundary collocation points for solving some problems for the Laplace operator

Boundary collocation is a method for obtaining approximate solutions of boundary problems for linear partial differential equations, for which complete families of particular solutions are explicitly known. The method contains various decisions which are important for its performance, such as choice of solution subspace, choice of basis for the subspace, and choice of collocation points. Using a model problem, some particular strategies for the determination of collocation points are investigated.     

Pseudospectral Legendre-based optimal computation of nonlinear constrained variational problems

It is well known that, spectrally accurate solution can be maintained if the grids on which a nonlinear physical problem is to be solved must be obtained by spectrally accurate techniques. In this paper, the pseudospectral Legendre method for general nonlinear smooth and nonsmooth constrained problems of the calculus of variations is studied. The technique is based on spectral collocation methods in which the trajectory, x(t), is approximated by the Nth degree interpolating polynomial, using Legendre-Gauss-Lobatto points as the collocation points, and Lagrange polynomials as trial functions. The integral involved in the formulation of the problem is approximated based on Legendre-Gauss-Lobatto integration rule, thereby reducing the problem to a nonlinear programming one to which existing well-developed algorithms may be applied. The method is easy to implement and yields very accurate results. Illustrative examples are included to confirm the convergence of the pseudospectral Legendre method. Moreover, a numerical experiment (on a nonsmooth problem) indicates that by applying a smoothing filter procedure to the pseudospectral Legendre approximation, one can recover the nonsmooth solution within spectral accuracy.     

ASYMPTOTIC ERROR EXPANSIONS OF QUADRATIC SPLINE COLLOCATION SOLUTIONS FOR TWO-POINT BOUNDARY VALUE PROBLEMS

In this paper, we consider the following problem:The quadratic spline collocation, with uniform mesh and the mid-knot points are taken as the collocation points for this problem is considered. With some assumptions, we have proved that the solution of the quadratic spline collocation for the nonlinear problem can be written as a series expansions in integer powers of the mesh-size parameter. This gives us a construction method for using Richardson's extrapolation. When we have a set of approximate solution with different mesh-size parameter a solution with high accuracy can he obtained by Richardson's extrapolation.     

A collocation method for solving singular integro-differential equations

This paper describes a collocation method for numerically solving Cauchy-type linear singular integro-differential equations. The numerical method is based on the transformation of the integro-differential equation into an integral equation, and then applying a collocation method to solve the latter. The collocation points are chosen as the Chebyshev nodes. Uniform convergence of the resulting method is then discussed. Numerical examples are presented and solved by the numerical techniques.     

An improved Chebyshev collocation algorithm for direct simulation of 2D turbulent convection in differentially heated cavities

We present a modified Chebyshev collocation algorithm for direct numerical simulations of 2D turbulent convection in differentially heated cavities. The numerical algorithm integrates the Navier-Stokes equations in velocity-pressure formulation with a Chebyshev spatial approximation and a second order finite difference time-stepping scheme. A coordinate stretching is introduced which allows one to redistribute the collocation points where needed in order to resolve more economically the small scales that appear at cavity mid-height. This algorithm is used to perform simulations in a square differentially heated cavity with adiabatic top and bottom walls filled with a fluid of Prandtl number equal to 0.71 for Rayleigh number values up to 10¹⁰ which is almost two orders of magnitude higher than the onset of unsteadiness. The time-dependent dynamics of the solutions are investigated and the time-averaged flow structures are displayed. The influence of unsteadiness on the local and global heat transfer coefficients is examined.     

Numerical solution of the problem of the stress-strain state in hollow cylinders using spline approximations

The three-dimensional theory of elasticity is used for a study of the stress-strain state in a hollow cylinder with varying stiffness. The corresponding problem is solved by a method that is partly analytical and partly numerical in nature: Spline approximations and collocation are used to reduce the partial differential equations of elasticity to a boundary-value problem for a system of ordinary differential equations of higher order for the radial coordinate, which is then solved using the method of stable discrete orthogonalization. Results for an inhomogeneous cylinder for various types of stiffness are presented.     

A moving collocation method for the solution of the transient convection–diffusion-reaction problems

An adaptive collocation method is introduced for simulating the short time diffusion–convection-reaction problem. The method is based on dividing the solution domain into active and inactive zones in such a way that the collocation points remain concentrated in regions of solution variation. The numerical results show that the proposed method is efficient in simulating the sharp profile at short time for the convective-dominant case. The adaptive scheme performance is found compatible with the high-order finite difference method, the QUICK method in terms of the CPU time and average numerical errors.     

Boundary collocation method for a cracked rectangular plate with double external tension

In this article, the boundary collocation method is employed to investigate the problems of a central crack in a rectangular plate which applied double external tension on the outer boundary under the assumption that the dimensions of the plate are much larger than that of the crack. A set of stress functions has also been proposed based on the theoretical analysis which satisfies the condition that there is no external force on the crack surfaces. It is only necessary to consider the condition on the external boundary. Using boundary collocation method, the linear algebra equations at collocation points are obtained. The least squares method is used to obtain the solution of the equations, so that the unknown coefficients can be obtained. According to the expression of the stress intensity factor at crack tip, we can obtain the numerical results of stress intensity factor. Numerical experiments show that the results coincide with the exact solution of the infinite plate. In particular, this case of the double external tension applied on the outer boundary is seldom studied by boundary collocation method.     

The boundary element spline collocation for nonuniform meshes on the torus

The spline collocation method for a class of biperiodic strongly elliptic pseudodifferential operators is considered. As trial functions tensor products of odd degree splines are used and the collocation is imposed at the nodal points of the tensor product mesh. It is shown that the collocation problem is uniquely solvable if the maximum mesh length is small enough. Moreover, the approximation is stable and quasioptimal with respect to a norm depending on the order of the operator and the degree of approximating splines. Some convergence results are given for general and quasiuniform meshes. The results cover for example the single layer and the hypersingular operators.