Performance Analysis of Error-Control B-spline Gaussian Collocation Software for PDEs

Pre-printB-spline Gaussian collocation software has been widely used in the numerical solution of boundary value ordinary differential equations (BVODEs) and partial differential equations (PDEs) in one space dimension (1D) for many years. The software package, BACOL, developed over a decade ago, was one of the first 1D PDE packages to provide both temporal and spatial error control. A new package, BACOLI, improves upon the efficiency of BACOL through the use of new types of spatial error estimation and control. The complexity of the interactions among the component numerical algorithms used by these packages implies that extensive testing and analysis of the test results is an essential factor in their development. In this paper, we investigate the performance of the BACOL and BACOLI packages with respect to several important machine independent algorithmic measures and examine the effectiveness of the new error estimation and error control strategies. We also investigate the influence of the choice of the degree of the B-splines on the efficiency and reliability of the solvers. These results will provide new insights into how to improve BACOLI, lead to improvements in the Gaussian collocation BVODE solvers, COLSYS and COLNEW, and guide the further development of B-spline Gaussian collocation software with error control for 2D PDEs

Matrix decomposition algorithms for the C(0)-quadratic finite element Galerkin method

Explicit expressions for the eigensystems of one-dimensional finite element Galerkin (FEG) matrices based on C-0 piecewise quadratic polynomials are determined. These eigensystems are then used in the formulation of fast direct methods, matrix decomposition algorithms (MDAs), for the solution of the FEG equations arising from the discretization of Poisson's equation on the unit square subject to several standard boundary conditions. The MDAs employ fast Fourier transforms and require O(N-2 log N) operations on an N x N uniform partition. Numerical results are presented to demonstrate the efficacy of these algorithms

An ADI extrapolated Crank-Nicolson orthogonal spline collocation method for nonlinear reaction-diffusion systems: a computational study

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 $O({\cal N})$ operations where ${\cal N}$ 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

An H<SUP>1</SUP>-Galerkin mixed finite element method for an evolution equation with a positive-type memory term

An H1-Galerkin mixed finite element method is analyzed for a class of evolution equations with memory. When a classical mixed method is applied to such problems, it has not been possible to obtain any estimate for the flux. However, the proposed approach yields optimal order convergence without the LBB consistency condition and quasi uniformity of the finite element mesh. Compared to the results proved for one space variable, the L2 estimate of the flux is not optimal for problems in two and three space dimensions. Therefore, a modification of the method is proposed and analyzed. A maximum norm estimate is also derived in one and two space variables. A backward Euler approximation of the modified method is also analyzed

Orthogonal Spline Collocation Methods For Schrödinger-Type Equations In One Space Variable

. We examine the use of orthogonal spline collocation for the semi-discretization of the cubic Schrodinger 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 Schrodinger equation, we present the results of numerical experiments in which the integration in time is performed using a routine from a software library. AMS(MOS) subject classifications. 65M15, 65M20, 65M70 1. Introduction. In this paper, we examine the use of orthogonal spline collocation, that is, spline collocation at Gauss points, for the semi-discretization of two problems of Schrodinger type. We first consider the initial value problem for the cubic Schrodinger equation iu t + u xx + qjuj 2 u = 0; (x; t) 2 (01; 1) 2 (0; T ]; u(x; 0) = g(x); x 2 (01; 1); (1.1) where i 2 = 01, q is a given positive constant, the given function g(x) is complexvalued, and jg(x)j ! 0 as jxj !1: This equat..

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 CrankNicolson and alternating direction implicit (ADI) schemes employing C 1 piecewise polynomial spaces of arbitrary degree 3 in each space variable. The stability of the schemes is examined and optimal order a priori L 2 - and H 1 -error estimates at each time step are derived. Parallel implementation of the ADI scheme is discussed. Key words. Schrodinger equation, orthogonal spline collocation, Gauss points, Crank-Nicolson scheme, alternating direction implicit method. AMS subject classifications. 65M70, 65M12, 65M15 1. Introduction. Orthogonal spline collocation (OSC) methods have proved to be exceedingly effective for the approximate solution of a broad class of problems (see, for example, [5, 9, 21, 28, 35, 36]). In comparison to finite difference methods, OSC provides approximations to the solu..