C. W. Gear

The automatic integration of ordinary differential equations

An integration technique for the automatic solution of an initial value problem for a set of ordinary differential equations is described. A criterion for the selection of the order of approximation is proposed. The objective of the criterion is to increase the step size so as to reduce solution time. An option permits the solution of “stiff” differential equations. A program embodying the techniques discussed appears in Algorithm 407.

Automatic integration of Euler-Lagrange equations with constraints☆

Abstract Numerical difficulties in the integration of Euler-Lagrange and similar equations are discussed. A technique for reducing their index from three to two is introduced and it is shown that variable-order, variable-step BDF methods converge for these index two problems. The practical application of this reduction in a numerical setting is examined.

A User’s View of Solving Stiff Ordinary Differential Equations

This paper aims to assist the person who needs to solve stiff ordinary differential equations.First we identify the problem area and the basic difficulty by responding to some fundamental questions: Why is it worthwhile to distinguish a special class of problems termed “stiff”? What are stiff problems? Where do they arise? How can we recognize them?Second we describe the characteristics shared by methods for the numerical solution of stiff problems. These characteristics have important implications as to the convenience and efficiency of solution of even routine problems. Understanding them is indispensable to the assembling of codes for the very efficient solution of special problems or for solving exceptionally large problems at all.Third we shall briefly discuss what is meant by “solving” a differential equation numerically and what might be reasonably expected in the case of stiff problems.

Multirate linear multistep methods

The design of a code which uses different stepsizes for different components of a system of ordinary differential equations is discussed. Methods are suggested which achieve moderate efficiency for problems having some components with a much slower rate of variation than others. Techniques for estimating errors in the different components are analyzed and applied to automatic stepsize and order control. Difficulties, absent from non-multirate methods, arise in the automatic selection of stepsizes, leading to a suggested organization of the code that is counter-intuitive. An experimental code and some initial experiments are described.

'Coarse' integration/bifurcation analysis via microscopic simulators: micro-Galerkin methods

Abstract We present a time-stepper based approach to the ‘coarse’ integration and stability/bifurcation analysis of distributed reacting system models. The methods we discuss are applicable to systems for which the traditional modeling approach through macroscopic evolution equations (usually partial differential equations, PDEs) is not possible because the PDEs are not available in closed form. If an alternative, microscopic (e.g. Monte Carlo or Lattice Boltzmann) description of the physics is available, we illustrate how this microscopic simulator can be enabled (through a computational superstructure) to perform certain integration and numerical bifurcation analysis tasks directly at the coarse, systems level. This approach, when successful, can circumvent the derivation of accurate, closed form, macroscopic PDE descriptions of the system. The direct ‘systems level’ analysis of microscopic process models, facilitated through such numerical ‘enabling technologies’, may, if practical, advance our understanding and use of nonequilibrium systems.

s -step iterative methods for symmetric linear systems

Abstract In this paper we introduce s -step Conjugate Gradient Method for Symmetric and Positive Definite (SPD) linear systems of equations and discuss its convergence. In the s -step Conjugate Gradient Method iteration s new directions are formed simultaneously from ≎ r i , Ar i ,…, A s −1 r i ≎ and the preceding s directions. All s directions are chosen to be A-orthogonal to the preceding s directions. The approximation to the solution is then advanced by minimizing an error functional simultaneously in all s directions. This intuitively means that the progress towards the solution in one iteration of the s -step method equals the progress made over s consecutive steps of the one-step method. This is proven to be true.

Iterative Solution of Linear Equations in ODE Codes

Each integration step of a stiff equation involves the solution of a nonlinear equation, usually by a quasi-Newton method which leads to a set of linear problems involving the Jacobian, J, of the differential equation. Iterative methods for these linear equations are studied. Of particular interest are methods which do not require an explicit Jacobian but can work directly with differences of function values using

Solving Ordinary Differential Equations with Discontinuities

J\delta \cong f(x + \delta ) - f(x)

THE NUMERICAL INTEGRATION OF ORDINARY DIFFERENTIAL EQUATIONS OF VARIOUS ORDERS

. Some numerical experiments using a modification of LSODE are reported

Approximation methods for the consistent initialization of differential-algebraic equations

An algorithm is described that can detect and locate some discontinuities and provide information about their size, order and position. However, the success of the algorithm is strongly dependent on the location of the discontinuity with respect to the steps that straddle it. The major advantage of the scheme appears to be that a more reliable error estimate can be used when a discontinuity is present so that codes will be more robust. In some cases significant savings may accrue but it appears that a better restarting procedure than the one used will be necessary to realize most of those benefits.