Orthogonal Spline Collocation Research Articles

On a discrete‐time collocation method for the nonlinear Schrödinger equation with wave operator

AbstractWe consider a discrete‐time orthogonal spline collocation scheme for solving Schrödinger equation with wave operator. The scheme is proposed recently by Wang et al. (J Comput Appl Math 235 (2011), 1993–2005) and is showed to have high‐order convergence rate when a parameter θ in the scheme is not less than \documentclass{article} \usepackage{amsmath,amsfonts, amssymb}\pagestyle{empty}\begin{document}$\frac{1}{4}$\end{document}. In this article, we show that the result can be extended to include \documentclass{article} \usepackage{amsmath,amsfonts, amssymb}\pagestyle{empty}\begin{document}$\theta\in(0,\frac{1}{4})$\end{document} under an assumption. Numerical example is given to justify the theoretical result. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

An ADI extrapolated Crank-Nicolson orthogonal spline collocation method for nonlinear reaction–diffusion systems

An alternating direction implicit (ADI) orthogonal spline collocation (OSC) method is described for the approximate solution of a class of nonlinear reaction–diffusion systems. Its efficacy is demonstrated on the solution of well-known examples of such systems, specifically the Brusselator, Gray-Scott, Gierer-Meinhardt and Schnakenberg models, and comparisons are made with other numerical techniques considered in the literature. The new ADI method is based on an extrapolated Crank-Nicolson OSC method and is algebraically linear. It is efficient, requiring at each time level only O(N) operations where N is the number of unknowns. Moreover, it is shown to produce approximations which are of optimal global accuracy in various norms, and to possess superconvergence properties.

Split-step orthogonal spline collocation methods for nonlinear Schrödinger equations in one, two, and three dimensions

Split-step orthogonal spline collocation (OSC) methods are proposed for one-, two-, and three-dimensional nonlinear Schrödinger (NLS) equations with time-dependent potentials. Firstly, the NLS equation is split into two nonlinear equations, and one or more one-dimensional linear equations. Commonly, the nonlinear subproblems could be integrated directly and accurately, but it fails when the time-dependent potential cannot be integrated exactly. In this case, we propose three approximations by using quadrature formulae, but the split order is not reduced. Discrete-time OSC schemes are applied for the linear subproblems. In numerical experiments, many tests are carried out to prove the reliability and efficiency of the split-step OSC (SSOSC) methods. Solitons in one, two, and three dimensions are well simulated, and conservative properties and convergence rates are demonstrated. We also apply the ways of solving the nonlinear subproblems to the split-step finite difference (SSFD) methods and the time-splitting spectral (TSSP) methods, and the approximate ways still work well. Finally, we apply the SSOSC methods to solve some problems of Bose–Einstein condensates.

Discrete-time orthogonal spline collocation methods for the nonlinear Schrödinger equation with wave operator

In this paper, discrete-time orthogonal spline collocation schemes are proposed for the nonlinear Schrödinger equation with wave operator. These schemes are constructed by using orthogonal spline collocation approaches combined with finite difference methods. The conservative property, the convergence, and the stability of these methods are theoretically analyzed and also verified by extensive numerical experiments. In addition, some interesting phenomena which require further theoretical analysis are discussed numerically.

ADI orthogonal spline collocation methods for parabolic partial integro-differential equations

Alternating direction implicit (ADI) orthogonal spline collocation schemes are formulated and analysed for a class of partial integro-differential equations of parabolic type. These techniques are based on the θ-method, for θ ∈ [1/2, 1], (where θ = 1 yields the backward Euler method and θ = 1/2 yields the Crank-Nicolson method) and the second-order backward differentiation formula (BDF) method. For each method, optimal estimates in various norms at each time step are derived and confirmed by results of numerical experiments.

An Alternating Direction Implicit Backward Differentiation Orthogonal Spline Collocation Method for Linear Variable Coefficient Parabolic Equations

We present a new fully discrete method to solve a linear variable coefficient parabolic equation on a rectangular region. The scheme is an alternating direction implicit method which uses the third order backward differentiation formula for the time discretization and piecewise Hermite bicubic orthogonal spline collocation for the spatial discretization. The $L^2$ norm stability and convergence analysis is carried out for the heat equation. The stability analysis reveals that the scheme is stable with respect to the right-hand sides but it is unstable with respect to the initial conditions. We explain, using a damping property of the scheme and round-off error analysis, why this instability is harmless. We also show how a different choice of initial values leads to optimal convergence orders. Numerical results demonstrate the third order accuracy in time and optimal order accuracy in space using maximum and Sobolev norms, respectively.

Matrix decomposition algorithms for the finite element Galerkin method with piecewise Hermite cubics

Matrix decomposition algorithms (MDAs) employing fast Fourier transforms are developed for the solution of the systems of linear algebraic equations arising when the finite element Galerkin method with piecewise Hermite bicubics is used to solve Poisson’s equation on the unit square. Like their orthogonal spline collocation counterparts, these MDAs, which require O(N2logN) operations on an N×N uniform partition, are based on knowledge of the solution of a generalized eigenvalue problem associated with the corresponding discretization of a two-point boundary value problem. The eigenvalues and eigenfunctions are determined for various choices of boundary conditions, and numerical results are presented to demonstrate the efficacy of the MDAs.

A class of conservative orthogonal spline collocation schemes for solving coupled Klein–Gordon–Schrödinger equations

A class of discrete-time orthogonal spline collocation schemes for solving coupled Klein–Gordon–Schrödinger equations with initial and boundary conditions are considered. These schemes are constructed by using piecewise cubic Hermite interpolations in space combined with finite difference methods in time. It is proved that the schemes have the conservation laws of discrete energy, and possess second order accuracy in maximum norm and fourth order in L 2 -norm for time and space, respectively. The conservation laws and rate of convergence are verified in numerical experiments. Moreover, the propagations of solitary waves and collisions of two head-on solitary waves are also well simulated.

Alternating Direction Implicit Orthogonal Spline Collocation Methods for an Evolution Equation with a Positive-Type Memory Term

New numerical techniques are presented for the solution of a class of linear partial integro-differential equations (PIDEs) with a positive-type memory term in the unit square. In these methods, orthogonal spline collocation (OSC) is used for the spatial discretization, and, for the time stepping, new alternating direction implicit (ADI) methods based on the backward Euler, the Crank-Nicolson, and the second order BDF methods combined with judiciously chosen quadrature rules are considered. The ADI OSC methods are proved to be of optimal accuracy in time and in the $L^2$ norm in space. Numerical results confirm the predicted convergence rates and also exhibit optimal accuracy in the $L^{\infty}$ and $H^1$norms and superconvergence phenomena.

Nonoverlapping domain decomposition with cross points for orthogonal spline collocation

Nonoverlapping domain decomposition with cross points for orthogonal spline collocation

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

AbstractA 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

Orthogonal spline collocation methods for the stream function‐vorticity formulation of the Navier–Stokes equations

AbstractAn orthogonal spline collocation (OSC) spatial discretization is proposed for the solution of the fully coupled stream function‐vorticity formulation of the Navier–Stokes equations in two dimensions. For the time‐stepping, a three‐level leapfrog scheme is employed. This method is algebraically linear, and, at each time step, gives rise to a system of linear equations of the form arising in the OSC approximation of the biharmonic Dirichlet problem and can be solved by a fast direct method. Error estimates in the Hl–norm in space, l = 1,2, are derived for the semi‐discrete method and the fully‐discrete leapfrog scheme which is also shown to be second order accurate in time. Numerical results are presented which confirm the theoretical analysis and exhibit superconvergence phenomena, which provide superconvergent approximations to the components of the velocity. © John Wiley & Sons, Inc. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008

Orthogonal cubic spline collocation method for the nonlinear parabolic equation arising in non-Newtonian fluid flow

Using the orthogonal cubic spline collocation method, solution for the nonlinear parabolic equation arising in magneto-hydrodynamic unsteady Poiseuille flow of the generalized Newtonian fluid (Carreau rheological model) is obtained. Also, using the Lyapunov functional, a bound for the maximum norm of the semi-discrete solution is derived. Moreover, optimal error estimates are established for the semi-discrete solution. Numerical results thus obtained are presented graphically and the salient features of the solution are discussed, for various values of the parameters. The results obtained reveal many interesting behaviors that warrant further study on the parabolic equations related to non-Newtonian fluid phenomena. Furthermore the analysis can be used to study the mathematical models that involve the flow of viscous fluids with shear rate-dependent properties: For example, models dealing with polymer processing, tribology and lubrication, and food processing.

Orthogonal cubic spline collocation method for the Cahn–Hilliard equation

The Cahn–Hilliard equation plays an important role in the phase separation in a binary mixture. This is a fourth order nonlinear partial differential equation. In this paper, we study the behaviour of the solution by using orthogonal cubic spline collocation method and derive optimal order error estimates. We discuss some computational experiments by using monomial basis functions in the spatial direction and RADAU 5 time integrator. The method we present here is better in terms of stability, efficiency and conditioning of the resulting matrix. Since no integrals to be evaluated or approximated, it behaves better than finite element method.

An error analysis and the mesh independence principle for a nonlinear collocation problem

AbstractA nonlinear Dirichlet boundary value problem is approximated by an orthogonal spline collocation scheme using piecewise Hermite bicubic functions. Existence, local uniqueness, and error analysis of the collocation solution and convergence of Newton's method are studied. The mesh independence principle for the collocation problem is proved and used to develop an efficient multilevel solution method. Simple techniques are applied for estimating certain discretization and iteration constants that are used in the formulation of a mesh refinement strategy and an efficient multilevel method. Several mesh refinement strategies for solving a test problem are compared numerically. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006

An Alternating-Direction Implicit Orthogonal Spline Collocation Scheme for Nonlinear Parabolic Problems on Rectangular Polygons

We consider a nonlinear parabolic initial-boundary value problem on a rectangular polygon with the solution satisfying Robin boundary conditions with variable coefficients. An approximation to the solution at the desired time value is obtained using an alternating-direction implicit extrapolated Crank--Nicolson scheme in which orthogonal spline collocation with piecewise polynomials of an arbitrary degree is used for spatial discretization. At each time level, the scheme determines the intermediate solution along horizontal lines and the approximate solution along vertical lines passing through the Gauss points. Only at the last time level is the approximate solution along vertical lines converted into the approximate solution defined on the entire rectangular polygon. This property of our approach leads to its efficient implementation and its applicability to rectangular polygons.

TH-collocation for the biharmonic equation

This paper is intended as a contribution to enhance orthogonal collocation methods. In this, a novel collocation method— TH-collocation—is applied to the biharmonic equation and the merits of such procedure are exhibited. TH-collocation relaxes the continuity requirements and, for the 2D problems here treated, leads to the development of algorithms for which the matrices are sparse (nine-diagonal), symmetric and positive definite. Due to these properties, the conjugate gradient method can be directly, and more effectively, applied to them. These features contrast with those of the standard orthogonal spline collocation on cubic Hermites, which yields matrices that are non-symmetric and non-positive. This paper is part of a line of research in which a general and unified theory of domain decomposition methods, proposed by Herrera, is being explored. Two kinds of contributions can be distinguished in this; some that are relevant for the parallel computation of continuous models and new discretization procedures for partial differential equations. The present paper belongs to this latter kind of contributions.

Multilevel Preconditioners for Non--self-adjoint or Indefinite Orthogonal Spline Collocation Problems

Efficient numerical algorithms are developed and analyzed that implement symmetric multilevel preconditioners for the solution of an orthogonal spline collocation (OSC) discretization of a Dirichlet boundary value problem with a non--self-adjoint or an indefinite operator. The OSC solution is sought in the Hermite space of piecewise bicubic polynomials. It is proved that the proposed additive and multiplicative preconditioners are uniformly spectrally equivalent to the operator of the normal OSC equation. The preconditioners are used with the preconditioned conjugate gradient method, and numerical results are presented that demonstrate their efficiency.

An orthogonal spline collocation alternating direction implicit method for second-order hyperbolic problems

On a rectangular region, we consider a linear second-order hyperbolic initial-boundary value problem involving a mixed derivative term, continuous variable coefficients and nonhomogeneous Dirichlet boundary conditions. In comparison to the alternating direction implicit Laplace-modified method of Fernandes (1997), we formulate and analyse a new parameter-free alternating direction implicit scheme in which the standard central difference formula is used for the time approximation and orthogonal spline collocation is used for the spatial discretization. We establish unconditional stability of the scheme, and its optimal order in the discrete maximum norm in time and the H 1 norm in space. Numerical experiments indicate that the new scheme, which has the same order as the method of Fernandes (1997, Numer. Math., 77, 223-241), is more accurate. We also show that the new scheme is easily generalized to the second-order hyperbolic problems on rectangular polygons. Extensions of the scheme to problems with discontinuous coefficients, nonlinear problems, and problems with other boundary conditions are also discussed.

A fast solver for the orthogonal spline collocation solution of the biharmonic Dirichlet problem on rectangles

A fast Schur complement algorithm is presented for computing the piecewise Hermite bicubic orthogonal spline collocation solution of the biharmonic Dirichlet problem on a rectangular region. On an N× N uniform partition, the algorithm, which involves the preconditioned conjugate gradient method and fast Fourier transforms, requires O( N 2log 2 N) arithmetic operations.