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

Linear system solvers for boundary value ODEs

AbstractWe investigate the stability properties of several linear system solvers for solving boundary value ODEs. We consider the compactification algorithm, Gaussian elimination with row partial pivoting, and a QR algorithm applied to linear systems arising from solving BVPs for which the matrix is block-bidiagonal except for bordering along the last n rows and columns. We will particularly compare AUTO's original linear solver (an LU decomposition with partial pivoting) and our implementation of the analogous QR algorithm to AUTO. Two other factors (the underlying continuation strategy and mesh selection strategy) may affect the stability of the linear system solver for ODE continuation codes as well and are also discussed in our numerical investigations

Extending BACOLI to solve multi-scale problems

The BACOLI package is a numerical software package for solving parabolic partial differential equations in one spatial dimension. It implements a B-spline collocation method for the spatial discretization of a system of partial differential equations. The resultant ordinary differential equations together with the boundary conditions form a system of differential-algebraic equations. The differential-algebraic equations are then solved using the DASSL solver. The BACOLI software package features adaptive error control in the temporal and spatial domains. The estimate of the temporal error is controlled through the DASSL solver. The estimate of the spatial error is controlled based on the difference between two solutions computed in the BACOLI software package. This difference gives an estimation of the error. If this error estimate does not meet the user-supplied tolerance, then the spatial mesh is changed. The BACOLI software package can only solve parabolic partial differential equations that depend on spatial derivatives. In this thesis, the BACOLI software package is modified to solve a broader spectrum of problems. In fact, after some modifications, the extended BACOLI software package can solve systems of parabolic partial differential equations and time-dependent equations that do not depend on spatial derivatives. We apply this extended software package to solve the monodomain model of cardiac electrophysiology. The monodomain model is a multi-scale mathematical model for the evolution of the electrical potential in cardiac tissue that couples the ionic currents at the cellular scale with their propagation at the tissue scale. Because of their local nature, the mathematical models of a single cell have no dependency on spatial derivatives whereas the models at the tissue level do. The heart models considered in our numerical experiments use various cardiac cell models. We find that solving the heart models through the extended BACOLI software package, in some cases, leads to a speed-up in comparison with the Chaste software package, which is a powerful, widely used, and well-respected software package for heart simulation

Asymptotically correct defect control software for boundary value ordinary differential equations

xii, 109 leaves : col. ill. ; 29 cm.Includes abstract.Includes bibliographical references (leaves 106-109).BVP_SOLVER II [Boisvert, Muir, Spiteri, 2013] is an efficient software package for the numerical solution of systems of boundary value ordinary differential equations. It employs discrete mono-implicit Runge-Kutta (MIRK) schemes to transform the ODEs into nonlinear systems which are solved by modified Newton iterations. Continuous MIRK interpolants then augment the discrete solutions from the nonlinear system, to obtain a continuous solution approximation across the problem domain. The code monitors solution quality through defect analysis and employs an adaptive mesh refinement strategy as a means of controlling the defect, which is the amount by which the computed solution fails to satisfy the ODEs. This thesis describes the development of new Hermite-Birkhoff interpolants and modifications to the BVP_SOLVER II software in order to implement a new defect estimation strategy called “Asymptotically Correct Maximum Defect Estimation”, based on the new interpolants. Numerical results which demonstrate the robustness and efficiency of the new strategy are presented

Spatial error estimation for collocation solutions of differential equations

1 online resource (117 pages) : illustrations (chiefly colour)Includes abstract and appendix.Includes bibliographical references (pages 72-74).Computational Science is now a central component of all scientific investigation, along with the traditional modes of experimental and theoretical investigation. Computational Science involves the development and solution of mathematical models, i.e., systems of equations, that represent approximations to real world phenomenon in a wide variety of scientific areas. These mathematical models typically do not have closed form solutions and thus the models are solved using computational software to obtain approximate solutions. Since these solutions are approximate, the question that must be addressed is “How Good is the Computed Solution?”. This question is answered for a given numerical solution through the computation of a good quality error estimate. This thesis will discuss current work on answering this question in the area of computational methods for differential equations that depend on time and/or one or more spatial dimensions. We will describe the use of collocation, a general numerical method that can be used to obtain approximate solutions for a wide range of problem classes, as well as our recent work in the development of efficiently computable error estimates for collocation solutions based on special types of interpolants. We provide results from numerical experiments to demonstrate the effectiveness of our approach

B-spline collocation for two dimensional, time-dependent, parabolic PDEs

vi, 177 leaves : ill. ; 29 cm.Includes abstract and appendices.Includes bibliographical references (leaves 82-88).In this thesis, we consider B-spline collocation algorithms for solving two-dimensional in space, time-dependent parabolic partial differential equations (PDEs), defined over a rectangular region. We propose two ways to solve the problem: (i) The Method of Surfaces: Discretizing the problem in one of the spatial domains, we obtain a system of one-dimensional parabolic PDEs, which is then solved using a one-dimensional PDE system solver. (ii) Two-dimensional B-spline collocation: The numerical solution is represented as a bi-variate piecewise polynomial with unknown time-dependent coefficients. These coefficients are determined by requiring the numerical solution to satisfy the PDE at a number of points within the spatial domain, i.e., we collocate simultaneously in both spatial dimensions. This leads to an approximation of the PDE by a large system of time-dependent differential algebraic equations (DAEs), which we then solve using a high quality DAE solver