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In the past few years, knowledge about methods for the numerical solution of two-point boundary value problems has increased significantly. Important theoretical and practical advances have been made in a number or fronts, although they are not adequately described in any tt'xt currently available. With this in mind, we organized an international workshop, devoted solely to this topic. Tht' workshop took place in Vancouver, B.C., Canada, in July 1()"13, 1984. This volume contains the refereed proceedings of the workshop. Contributions to the workshop were in two formats. There were a small number of invited talks (ten of which are presented in this proceedings); the other contributions were in the rorm or poster sessions, for which there was no parallel activity in the workshop. We had attemptt'd to cover a number of topics and objectives in the talks. As a result, the general review papt'rs of O'Malley and Russell are intended to take a broader perspective, while the other papers are more specific. The contributions in this volume are divided (somewhat arbitrarily) into five groups. The first group concerns fundamental issues like conditioning and decoupling, which have only rect'ntly gained a proper appreciation of their centrality. Understanding of certain aspects or shooting methods ties in with these fundamental concepts. The papers of Russell, dt' Hoog and Mattheij all deal with these issues.



Conditioning, dichotomy and related numerical considerations

A Unified View of Some Recent Developments in the Numerical Solution of Bvodes

At present it is not an easy task to learn the theory underlying the numerical methods for solving BVPs (boundary value problems) for ODEs, There has not been a single book published which treats the basic methods since the outstanding book by Keller [2] in 1968. There have been few meetings on this topic (although see [3,4,5,6]). One of the purposes of this paper is to briefly discuss numerical methods which have proved useful in practice.
Robert D. Russell

The Role of Conditioning in Shooting Techniques

This paper examines shooting and multiple shooting as a technique for the analysis of numerical schemes applied to two point boundary value problems. The aim of the analysis is to deduce convergence of a numerical scheme by establishing the convergence of the scheme when applied to a number of subproblems. Since such an approach is useful only if the subproblems are reasonably well conditioned, the question of conditioning is addressed. It is shown that there exist subproblems that are at least as well conditioned as the original problem. Two examples are presented for which the shooting approach leads to a substantial simplification in the analysis.
Frank de Hoog, Robert Mattheij

On Non-Invertible Boundary Value Problems

For non-invertible boundary value problems, i.e. where the boundary conditions as such do not determine the solution uniquely, the usual concepts of condition numbers and stability do not apply. Such problems typically arise when the interval is semi-infinite. If one assumes that the desired solution is bounded the boundary conditions are sufficient to give a unique solution (as an element of the bounded solutions manifold). Another type of problems are eigenvalue problems, where both the dynamics and the boundary conditions are homogeneous. We shall introduce sub-condition numbers that indicate the sensitivity of the problem with respect to perturbations of a relevant subproblem. We also discuss a numerical method that computes such sub-condition number to demonstrate its applicability. Finally we give a number of numerical examples to illustrate both the theory and the computational method.
R. M. M. Mattheij, F. R. de Hoog

Riccati Transformations: When and How to Use?

In this paper the problem of interest is a well-conditioned n-dimensional boundary value problem (BVP):
$$ \begin{array}{*{20}{c}} {\mathop x\limits^ \bullet \left( t \right) = A\left( t \right)x\left( t \right) + f\left( t \right)} {,t \in \left( {0,1} \right)} \end{array}, $$
subject to the boundary conditions
$$ {{B}^{0}}x(0) = {{B}^{1}}x\left( 1 \right) = b $$
(B0,B1 ∈IRn×n and b ∈ IRn).
Paul van Loon

Discretizations with Dichotomic Stability for Two-Point Boundary Value Problems

For a two-point boundary value problem to be well conditioned, the system of ordinary differential equations must necessarily possess a dichotomic set of fundamental solutions [7], with decaying modes controlled by initial conditions, and growing modes controlled by terminal conditions [10]. It was shown in [3] that it is important for a discretization of such a problem to preserve the dichotomy property, and the implications of this stability criterion were examined in a number of particular cases. Some simple difference schemes were examined for second order ordinary differential equations, and also various discretizations for first order systems, including those obtained by piecewise collocation and implicit Runge-Kutta type formulae. The last two examples were of multistep schemes, of such a form that they could be used in a sequential stepping mode, as would be done in shooting methods, or more generally in multiple shooting.
Roland England, Robert M. M. Mattheij

Implementation aspects of various methods

Improving the Performance of Numerical Methods for Two Point Boundary Value Problems

We report on an ongoing investigation into the performance of numerical methods for two point boundary value problems. We outline how methods based on multiple shooting, collocation and other local discretizations can share a common structure. The identification of this common structure permits us to analyse how various components of a method interact and also permits us to consider the assembly of a collection of modular routines which will eventually form the basis for a software environment for solving two point boundary value problems.
The initial stage of our investigation has involved the implementation and analysis of a family of multiple shooting methods as well as a family of collocation/Runge-Kutta methods. We have analysed the performance of these methods on a class of singular perturbation problems. The numerical conditioning of both families of methods on such problems and the convergence requirements of the corresponding iteration schemes (used to solve the discretized problem) has been investigated. Appropriate modifications to these methods which permit the effective solution of such problems will be discussed. We will also identify a subfamily of the Runge-Kutta methods that are particularly effective.
W. H. Enright

Reducing the Number of Variational Equations in the Implementation of Multiple Shooting

The standard method of multiple shooting for a system of n first order differential equations, with k unknown initial conditions requires the integration of k sets of variational equations on the first shot, and n sets of variational equations on every shot thereafter. This paper describes a variant of multiple shooting that requires the solution of k sets of variational equations on every shot. The technique applies to both linear and nonlinear boundary value problems. Techniques to deal with difficulties unique to the solution of nonlinear problems are suggested.
Fred T. Krogh, J. P. Keener, Wayne H. Enright

The Spline-Collocation and the Spline-Galerkin Methods for Orr-Sommerfeld Problem

Recently the projectional mesh methods of solving the boundary value problems for ordinary differential equa­tions are intensively developing. The theoretical and prac­tical aspects of the spline-collocation method have been examined by R.D.Russell and L.F.Shampine [15] . C. de Boor and B.Swartz [5] have studied a question of choosing the points in the spline-collocation method. They have shown that the greatest rate will be achieved by the choice of the Gaussian points as the collocation points. B.P.Kolobov and A.G.Sleptsov [11,12] have made use of the following method for solving the problem
$$ Ly = f,\left( {{B_o}y} \right)\left( a \right) = 0,\left( {{B_1}y} \right)\left( b \right) = 0 $$
where L is linear mth order differential operator. Let x1=a < x2 <… < xN+1=b, n > 0 is some interger. For each k=1, … , N−m the approximate solution will be a polynomial of the degree m+n−1 on the segment [xk,xk+m] . m coefficients of this polynomial are determined by the conditions uk(xi)=vi, i=k, … , k+m−1, the rest of the coefficients are determined by the collocation equations.The equalities vk+m=uk(xk+m), k=1, … , N−m, give N−m linear equations to determine the N unknowns vi. The boundary conditions yield m more equations.
A. G. Sleptsov

Singular perturbation (‘stiff’) problems

On the Simultaneous Use of Asymptotic and Numerical Methods to Solve Nonlinear Two Point Problems with Boundary and Interior Layers

The purpose of this paper is to provide a broad-brush survey concerning boundary value problems for certain systems of nonlinear singularly perturbed ordinary differential equations. The aim is to emphasize important and difficult open problems needing much more study, in terms of both mathematical and numerical analysis and computational experiments. The presentation will, regrettably, be removed from both direct applications and substantial achievements. Gradually, we will, however, become more specific and ultimately will discuss some currently tractible problems.
Robert E. O’Malley

Two Families of Symmetric Difference Schemes for Singular Perturbation Problems

Singularly perturbed boundary value ordinary differential problems are considered. For numerical approximation, families of symmetric difference schemes, which are equivalent to certain collocation schemes based on Gauss and Lobatto points, are used. The performance of these two families of schemes is compared. while Lobatto schemes are more accurate for some classes of problems, Gauss schemes are more stable in general.
Uri Ascher

A Numerical Method for Singular Perturbation Problems with Turning Points

Until recently, the development of general numerical methods for singular perturbation problems whose solutions exhibit internal layer type behavior has been largely neglected. Indeed, even the analytic study of general systems of first order ODEs with this type of behavior appears to be quite limited; perhaps this is one of the reasons for the lack of progress in this area. In this paper, we report on results obtained with H. O. Kreiss and N. Nichols [7], which address this problem. We will give a presentation which is somewhat different than that in [7], with the hope of emphasizing the similarity of our approach to the ideas that underlie the analytic technique of matched asymptotic expansions. We also present some recent results for nonlinear problems with internal layer behavior which have been obtained together with W. L. Kath and H. O. Kreiss. All of our results are for the two-point boundary value problem for systems of first-order ordinary differential equations. Since the solutions of singular perturbation problems of this type typically vary on two or more scales, these problems are often called “stiff” boundary value problems as well.
David L. Brown

Numerical solution of singular perturbed boundary value problems using a collocation method with tension splines

A collocation method is given for the numerical solution of a singular perturbed boundary value problem of the form:
$$ \begin{gathered} y' = f\left( {x,y,\varepsilon } \right){\kern 1pt} a \leqslant x \leqslant b \hfill \\ r\left( {y\left( a \right),y\left( b \right)} \right) = 0 \hfill \\ y = {\left( {{y_1}, \ldots ,{y_n}} \right)^t}{\kern 1pt} {y_i}:\left[ {a,b} \right] \to R \hfill \\ f = {\left( {{f_1}, \ldots ,{f_n}} \right)^t}{\kern 1pt} {f_i}:\left[ {a,b} \right] \to R \hfill \\ r = {\left( {{r_1}, \ldots ,{r_n}} \right)^t}{\kern 1pt} {r_i}:R \times R \to R \hfill \\ \end{gathered} $$
Maximilian R. Maier

Bifurcation problems and delay differential equations

Solving Boundary Value Problems for Functional Differential Equations by Collation

In many fields of application, such as Chemistry or Biology, processes appear which are more naturally modeled by functional differential equations (FDE’s) than by ordinary differential equations (ODE’s). Boundary value problems (BVP’s) for FDE’s appear frequently in the context of optimal control problems.
G. Bader

The Approximation of Simple Singularities

Many physical problems can be formulated as a parameter dependent nonlinear operator equation
$$ f(z,\lambda ) = 0,f:D \subset Z \times R \to Y $$
where Z and Y are Banach spaces and D is an open set in Z × R. Even if the original problem does not contain any explicit “control” parameters such as λ in (1.1), it may be advantageous to relax some physical constant, e.g., viscosity, or introduce an artificial parameter. By then analyzing (1.1) for certain critical or special values of λ, one may be able to characterize the full solution set
$$ {M_\lambda } \equiv \{ z \in Z:f(z,\lambda ) = 0\} $$
for any given value of λ. Whenever the Frechet derivative fz has range all of Y, the slices Mλ form locally smooth manifolds of dimension
$$ i \equiv null({f_z}) - def({f_z}) $$
and vary smoothly in λ. Here i denotes the general Fredholm index of fz, which we assume to be constant and nonnegative in the domain D. Of particular interest is the square case i= 0, in which the solutions z = z(λ) ∈ Mλ are locally unique and differentiable in λ as long as fz(z(λ),λ) possesses a bounded inverse. This observation is the basis of continuation or homotopy methods [14].
A. Griewank, G. W. Reddien

Calculating the Loss of Stability by Transient Methods, with Application to Parabolic Partial Differential Equations

The calculation of Hopf bifurcations in systems of parabolic PDEs (one space variable) is considered. By semidiscretization via method of lines one obtains ODE systems, thereby enabling the usage of ODE methods. Some novel results are presented dealing with transient methods, i.e. methods that handle the steady state as a special periodic solution with amplidude zero. The results include the calculation of stability of periodic orbits as well as the computation of points of loss of stability.
R. Seydel

A Runge-Kutta-Nystrom Method for Delay Differential Equations

Let consider the following boundary value problem for second order delay differential systems:
$$ \begin{gathered} y''\left( t \right) = f\left( {t,y\left( t \right),y'\left( t \right),y\left( {t - \tau \left( t \right)} \right),y'\left( {t - \sigma \left( t \right)} \right)} \right) {t_o} \leqslant t \leqslant b \hfill \\ y'\left( t \right) = \phi \left( t \right) t \leqslant {t_o} \hfill \\ y'\left( t \right) = \phi '\left( t \right) t < {t_o} \hfill \\ y\left( b \right) = {y_b} \hfill \\ \end{gathered} $$
y: IR → IRm, f: [ to,b ]×IR4m → IR and τ(t), σ(t)>0.
A. Bellen

Special applications

A Finite Difference Method for the Basic Stationary Semiconductor Device Equations

In this paper we analyse a special-purpose finite difference scheme for the basic stationary semiconductor device equations in one space dimension. These equations model potential distribution, carrier concentration and current flow in an arbitrary one-dimensional semiconductor device and they consist of three second order ordinary differential equations subject to boundary conditions. A small parameter appears as multiplier of the second derivative of the potential, thus the problem is singularly perturbed. We demonstrate the occurence of internal layers at so called device-junctions, which are jump-discontinuities of the data, and present a finite difference scheme which allows for the resolution of these internal layers without employing an exceedingly large number of grid-points. We establish the relation of this scheme to exponentially fitted schemes and give a convergence proof. Moreover the construction of efficient grids is discussed.
Peter A. Markowich

Solution of Premixed and Counterflow Diffusion Flame Problems by Adaptive Boundary Value Methods

Combustion models that simulate pollutant formation and study chemically controlled extinction limits in flames often combine detailed chemical kinetics with complicated transport phenomena. Two of the simplest models in which these processes are studied are the premixed laminar flame and the counterflow diffusion flame. In both cases the flow is essentially one-dimensional and the governing equations can be reduced to a set of coupled nonlinear two-point boundary value problems with separated boundary conditions.
Mitchell D. Smooke, James A. Miller, Robert J. Kee


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