Control of step size and order in extrapolation codes

AbstractExtrapolation of the semi-implicit midpoint rule is an effective way to solve stiff initial value problems for a system of ordinary differential equations. The theory of the control of step size and order is advanced by investigating questions not taken up before, providing additional justification for some algorithms, and proposing an alternative to the information theory approach of Deuflhard. An experimental code SIMP implementing the algorithms proposed is shown to be as good as, and in some respects better than, the research code METAN1 of Bader and Deuflhard

Variable order Adams codes

AbstractVariable step size, variable order (VSVO) Adams codes are very effective for solving initial value problems for first-order systems of ordinary differential equations. The theory of fixed-order codes is classical, but when the order is varied, there is no theory explaining fundamental issues. With realistic assumptions about order and step size selection, we prove convergence, approximate locally the behavior of the error, and justify standard error estimators

Embedded Eigenvalues and the Nonlinear Schrodinger Equation

A common challenge to proving asymptotic stability of solitary waves is understanding the spectrum of the operator associated with the linearized flow. The existence of eigenvalues can inhibit the dispersive estimates key to proving stability. Following the work of Marzuola & Simpson, we prove the absence of embedded eigenvalues for a collection of nonlinear Schrodinger equations, including some one and three dimensional supercritical equations, and the three dimensional cubic-quintic equation. Our results also rule out nonzero eigenvalues within the spectral gap and, in 3D, endpoint resonances. The proof is computer assisted as it depends on the sign of certain inner products which do not readily admit analytic representations. Our source code is available for verification at http://www.math.toronto.edu/simpson/files/spec_prop_asad_simpson_code.zip.Comment: 29 pages, 27 figures: fixed a typo in an equation from the previous version, and added two equations to clarif

An efficient Runge-Kutta (4,5) pair

AbstractA pair of explicit Runge-Kutta formulas of orders 4 and 5 is derived. It is significantly more efficient than the Fehlberg and Dormand-Prince pairs, and by standard measures it is of at least as high quality. There are two independent estimates of the local error. The local error of the interpolant is, to leading order, a problem-independent function of the local error at the end of the step

Delay-Differential Equations with Constant Lags

This article concerns delay-differential equations (DDEs) with constant lags. DDEs increasingly are being used to model various phenomena in mathematics and the physical sciences. For such equations the value of the derivative at any time depends on the solution at a previous lagged time. Although solving DDEs is similar in some respects to solving ordinary differential equations (ODEs), it differs in some rather significant ways. These differences are discussed briefly. The effect the differences can have on systems of ODEs and DDEs is illustrated. Popular approaches used in the development of numerical methods for solving DDEs are described. Available Matlab DDE solvers and a Fortran 90 solver based on these approaches are mentioned. Finally, some pointers to further resources available to interested readers are given

An Efficient Runge-Kutta (4,5) pair

A pair of explicit Runge-Kutta formulas of orders 4 and 5 is derived. It is significantly more efficient than the Fehlberg and Dormand-Prince pairs, and by standard measures it is of at least as high quality. There are two independent estimates of the local error. The local error of the interpolant is, to leading order, a problem-independent function of the local error at the end of the step

RAPID analysis of variable stiffness beams and plates: Legendre polynomial triple-product formulation

Copyright © 2017 John Wiley & Sons, Ltd. Numerical integration techniques are commonly employed to formulate the system matrices encountered in the analysis of variable stiffness beams and plates using a Ritz based approach. Computing these integrals accurately is often computationally costly. Herein, a novel alternative is presented, the Recursive Analytical Polynomial Integral Definition (RAPID) formulation. The RAPID formulation offers a significant improvement in the speed of analysis, achieved by reducing the number of numerical integrations that are performed by an order of magnitude. A common Legendre Polynomial basis is employed for both trial functions and stiffness/load variations leading to a common form for the integrals encountered. The Legendre Polynomial basis possesses algebraic recursion relations that allow these integrals to be reformulated as triple-products with known analytical solutions, defined compactly using the Wigner (3j) coefficient. The satisfaction of boundary conditions, calculation of derivatives and transformation to other bases is achieved through combinations of matrix multiplication, with each matrix representing a unique boundary condition or physical effect, therefore permitting application of the RAPID approach to a variety of problems. Indicative performance studies demonstrate the advantage of the RAPID formulation when compared to direct analysis using Matlab's ‘integral’ and ‘integral2’. Copyright © 2017 John Wiley & Sons, Ltd