Orthogonal Spline Collocation Research Articles

A Nonoverlapping Domain Decomposition Method for Orthogonal Spline Collocation Problems

Anonoverlapping domain decomposition approach is used on uniform and matching grids to first define and then to compute the orthogonal spline collocation solution of the Dirichlet boundary value problem for Poisson's equation on an L-shaped region. We prove existence and uniqueness of the collocation solution and derive optimal order Hs-norm error bounds for s=0,1,2$. The collocation solution on two interfaces is computed using the preconditioned conjugate gradient method, and the collocation solution on three squares is computed by a matrix decomposition method that uses fast Fourier transforms. The total cost of the algorithm is O(N2 \log N), where the number of unknowns in the collocation solution is O(N2 )$.

A Preconditioned Conjugate Gradient Method for Nonselfadjoint or Indefinite Orthogonal Spline Collocation Problems

We study the computation of the orthogonal spline collocation solution of a linear Dirichlet boundary value problem with a nonselfadjoint or an indefinite operator of the form $Lu=\sum a_{ij}(x)u_{x_ix_j}+\sum b_i(x) u_{x_i}+c(x)u$. We apply a preconditioned conjugate gradient method to the normal system of collocation equations with a preconditioner associated with a separable operator, and prove that the resulting algorithm has a convergence rate independent of the partition step size. We solve a problem with the preconditioner using an efficient direct matrix decomposition algorithm. On a uniform N × N partition, the cost of the algorithm for computing the collocation solution within a tolerance $\epsilon$ is O$(N^{2}\ln N |\ln \epsilon|)$.

Discrete-Time Orthogonal Spline Collocation Methods for Vibration Problems

Discrete-time orthogonal spline collocation schemes are formulated and analyzed for vibration problems involving various boundary conditions. Each problem is written as a Schrödinger-type system, which is then approximated by Crank--Nicolson and/or alternating direction implicit orthogonal spline collocation schemes. These schemes are shown to be second-order accurate in time and of optimal order accuracy in space in the Hm-norm, m=1,2.

Orthogonal spline collocation methods for partial differential equations

This paper provides an overview of the formulation, analysis and implementation of orthogonal spline collocation (OSC), also known as spline collocation at Gauss points, for the numerical solution of partial differential equations in two space variables. Advances in the OSC theory for elliptic boundary value problems are discussed, and direct and iterative methods for the solution of the OSC equations examined. The use of OSC methods in the solution of initial–boundary value problems for parabolic, hyperbolic and Schrödinger-type systems is described, with emphasis on alternating direction implicit methods. The OSC solution of parabolic and hyperbolic partial integro-differential equations is also mentioned. Finally, recent applications of a second spline collocation method, modified spline collocation, are outlined.

A Crank–Nicolson orthogonal spline collocation method for vibration problems

A discrete-time orthogonal spline collocation scheme is formulated and analyzed for a problem governing the transverse vibrations of a clamped square plate. The problem is reformulated as a Schrödinger-type system which is then approximated by a Crank–Nicolson orthogonal spline collocation scheme. This scheme is shown to be second-order accurate in time and of optimal order accuracy in space in the H 1 - and H 2 -norms.

Orthogonal Spline Collocation for Nonlinear Dirichlet Problems

We study the orthogonal spline collocation (OSC) solution of a homogeneous Dirichlet boundary value problem in a rectangle for a general nonlinear elliptic partial differential equation. The approximate solution is sought in the space of Hermite bicubic splines. We prove local existence and uniqueness of the OSC solution, obtain optimal order H1 and H2 error estimates, and prove the quadratic convergence of Newton's method for solving the OSC problem.

Superconvergence of the orthogonal spline collocation solution of Poisson's equation

Superconvergence of the orthogonal spline collocation solution of Poisson's equation

Superconvergence of the orthogonal spline collocation solution of Poisson's equation

Superconvergence of the orthogonal spline collocation solution of Poisson's equation

An Orthogonal Spline Collocation Alternating Direction Implicit Crank--Nicolson Method for Linear Parabolic Problems on Rectangles

We formulate an alternating direction implicit Crank--Nicolson scheme for solving a general linear variable coefficient parabolic problem in nondivergence form on a rectangle with the solution subject to nonhomogeneous Dirichlet boundary condition. Orthogonal spline collocation with piecewise Hermite bicubics is used for spatial discretization. We show that for sufficiently small time stepsize the scheme is stable and of optimal-order accuracy in time and the H1 norm in space. We also describe an efficient implementation of the scheme and present numerical results demonstrating the accuracy and convergence rates in various norms.

Orthogonal spline collocation methods for biharmonic problems

Orthogonal spline collocation methods are formulated and analyzed for the solution of certain biharmonic problems in the unit square. Particular attention is given to the Dirichlet biharmonic problem which is solved using capacitance matrix techniques.

Convergence Analysis of Orthogonal Spline Collocation for Elliptic Boundary Value Problems

Existence, uniqueness, and optimal order H2 , H1 , and L2 error bounds are established for the orthogonal spline collocation solution of a Dirichlet boundary value problem on the unit square. The linear, elliptic, nonself-adjoint, partial differential equation is given in nondivergence form. The approximate solution, which is a tensor product of continuously differentiable piecewise polynomials of degree $r\geq 3$, is determined by satisfying the partial differential equation at the nodes of a composite Gauss quadrature.

Discrete-time Orthogonal Spline Collocation Methods for Schrödinger Equations in Two Space Variables

Two discrete-time orthogonal spline collocation schemes are formulated and analyzed for solving the linear time-dependent Schrodinger equation in two space variables. These are Crank--Nicolson and alternating direction implicit (ADI) schemes employing C1 piecewise polynomial spaces of arbitrary degree $\ge 3$ in each space variable. The stability of the schemes is examined and optimal order a priori L2- and H1-error estimates at each time step are derived. Parallel implementation of the ADI scheme is discussed.

Efficient orthogonal spline collocation methods for solving linear second order hyperbolic problems on rectangles

Piecewise Hermite bicubic orthogonal spline collocation Laplace-modified and alternating-direction schemes for the approximate solution of linear second order hyperbolic problems on rectangles are analyzed. The schemes are shown to be unconditionally stable and of optimal order accuracy in the \(H^1\) and discrete maximum norms for space and time, respectively. Implementations of the schemes are discussed and numerical results presented which demonstrate the accuracy and rate of convergence using various norms.

Multilevel additive and multiplicative methods for orthogonal spline collocation problems

Multilevel preconditioners are proposed for the iterative solution of the discrete problems which arise when orthogonal spline collocation with piecewise Hermite bicubics is applied to the Dirichlet boundary value problem for a self-adjoint elliptic partial differential equation on a rectangle. Additive and multiplicative preconditioners are defined respectively as sums and products of independent operators on a sequence of nested subspaces of the fine partition approximation space. A general theory of additive and multiplicative Schwarz methods is used to prove that the preconditioners are spectrally equivalent to the collocation discretization of the Laplacian with the spectral constants independent of the fine partition stepsize and the number of levels. The preconditioned conjugate gradient and preconditioned Orthomin methods are considered for the solution of collocation problems. An implementation of the methods is discussed and the results of numerical experiments are presented.

The solution of nonlinear Schrödinger equations using orthogonal spline collocation

An orthogonal spline collocation semidiscretization, previously applied in the numerical solution of the cubic Schrödinger equation, is extended to the solution of Schrödinger equations which possess general power nonlinearities and nonlinear terms that contain derivatives. The accuracy of the method is examined, as well as its ability to conserve approximations to certain invariants associated with the theoretical solution. The results of numerical experiments are presented in which the integration in time is performed using a routine from a software library. Particular attention is paid to the value and limitations of conservation as an indicator of accuracy.

Iterative Algorithms for Orthogonal Spline Collocation Linear Systems

In this paper, we present several block iterative algorithms for solving the linear systems which arise from the discretization of second-order separable elliptic partial differential equations with Hermite cubic collocation. The convergence rates of these algorithms are estimated.

Matrix Decomposition Algorithms in Orthogonal Spline Collocation for Separable Elliptic Boundary Value Problems

Fast direct methods are presented for the solution of linear systems arising in high-order, tensor-product orthogonal spline collocation applied to separable, second order, linear, elliptic partial...

Fourier Matrix Decomposition Methods for the Least Squares Solution of Singular Neumann and Periodic Hermite Bicubic Collocation Problems

The use of orthogonal spline collocation with piecewise Hermite bicubics is examined for the solution of Poisson’s equation on a rectangle subject to either pure Neumann or pure periodic boundary conditions. Emphasis is placed on finding a least squares solution of these singular collocation problems. The technique of matrix decomposition is applied and explicit formulas for the requisite eigensystems corresponding to two-point Neumann and periodic collocation boundary value problems are presented. The resulting algorithms use fast Fourier transforms for efficiency and are highly parallel in nature. On an $N \times N$ partition, a fourth order accurate least squares solution is computed at a cost of $O(N^2 \log N)$ operations. The results of numerical experiments are provided that demonstrate that the implementations compare very favorably with recent fourth order accurate finite difference and finite element Galerkin codes.

A generalized analysis of internal diffusion of immobilized enzyme in multi-enzyme reactions

A study of the influence of diffusional limitations on the kinetic behavior of consecutive immobilized enzymatic reactions is presented. The consecutive sequence of reactions takes place inside a permeable particle and the system is analysed for the cases in which each enzyme catalysed step in the reaction scheme follows Michaelis-Menten type kinetics, with and without inhibition by its substrate or by its product and with a reversible mechanism. The theoretical analysis uses the orthogonal spline collocation method to obtain the effectiveness factors of the reaction sequence. It is shown that the reaction system exhibits the characteristics of either reaction control or diffusion control depending on the operating conditions.

Orthogonal spline collocation methods for Schr\"{o}dinger-type equations in one space variable

We examine the use of orthogonal spline collocation for the semi-discreti\-za\-tion of the cubic Schr\"{o}dinger equation and the two-dimensional parabolic equation of Tappert. In each case, an optimal order\(L^2\) estimate of the error in the semidiscrete approximation is derived. For the cubic Schr\"{o}dinger equation, we present the results of numerical experiments in which the integration in time is performed using a routine from a software library.