Superconvergence and the computation of generalized turning points for B.V.P’s

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Abstract

We consider projection methods in general and collocation at gauss points in particular for the numerical computation of generalized turning points for two point B.V.P’s. We discretize the original problem using collocation to obtain a finite dimensional problem. We will employ a direct method for the characterization and computation of simple turning points. Proper extensions of this direct method will be used for the computation of bifurcation points and cubic turning points. We comment also on the superconvergence which will be clear from the numerical examples.

Introduction

We will consider the numerical implementation of the accurate location of generalized turning points for two point boundary value problems. Such problem has received much attention recently. It is a well known fact that the application of conventional numerical methods leads to difficulties near such singularities, see for example, the proceedings edited by Kupper, Mittelman and Weber [10], Keller and Antman [7], Keller [8] and the review by Stakgold [13]. The purpose of this paper is to present a general discussion on the projection methods applied to such singular points and in particular to consider the application of collocation at gauss points, to give an idea about the systems to be solved and to show that collocation proved to be highly accurate and superconverges to the singular point. Related work on the discretization can be found in, for example, Attili [1], [4] where finite differences was used, Weber [14] had used multiple shooting for bifurcation problems only, Brezzi, Rappaz and Raviart [5] considered Rayleigh-Ritz method for simple turning points and Kikuchi [9] used finite elements. The problem under consideration is of the formF(y,λ)=0,where F:Rn × R  Rn ; and F is a C3–function. Here F(y, λ) = 0 is the finite dimensional discretization of the two point B.V.P through, say, the use of collocation. We assume the codimension of Range(Fy) is no more than one on Rn and Fλ  Range(Fy) when the codim. = 1. It follows then (Attili [1], [3]) that the solution manifold F−1(0) is a p-dimensional (p = 1) and the singular manifold is an n + 1  p dimensional. As a result, N(Fy) = Span{ϕ0} and Range(Fy)={yRn;ψ0Ty=0}. To characterize the singular manifold, choose T, r  Rn so that the matrixA=FyrTT0,is nonsingular near (y0, λ0) the singular point. This can be the case at the simple turning point if we choose r  Range(Fy) and TTϕ0 = 1. Next we solve the systemsAvg=01,uTgT·A=(0,1)for u,v and g.

Lemma 1

Both systems in (1.3) are uniquely solvable and (a) g = uTFyv, (b) g=-uTFyv-(uTrgT+gTTTv), where the prime denotes differentiation in y and λ.

The proof is straight forward and can be seen from the discussion presented before the statement of the lemma. Note that in applying this procedure, we normally choose r and TT to be constants which reduces (b) of the lemma to gy = uTFyyv and gλ = uTFv. Now to numerically compute the generalized turning point, we solve the extended systemF(y,λ)=0,g(y,λ)=0.

Note also that solving this system means finding the intersection between the solution manifold represented by F(y, λ) = 0 and the singular manifold represented by g(y, λ) = 0. Newtons method can be used to solve the system in (4). To find the gradient of g, we use the formulagyuT(Fy(y+ϵv,λ)-Fy(y,λ))/ϵ,which is based on the exact formula for derivative of g. The system in (4) is regular if and only ifyg(y0,λ0)v,is of full rank, see Attili [1], [3].

To consider the bifurcation case, we will assume Fλ  Range(Fy). As a result N(Fy, Fλ) will be spanned by two vectors and hence we will have the following lemma

Lemma 2

Let Fy˜=(Fy,Fλ);y˜=(y,λ) and if Fy has a one dimensional null space spanned by ϕ0 and ψ0TFλ=0. Then N(Fy) =  Span{Φ0, Φ1} where Φ0 = 0, 0) and Φ1=(yˆ,γ) where yˆRange(Fy) and γ  R such that Fyyˆ+γFλ=0.

This lemma means that v and g will each have two components. In general, in the case of bifurcation the system to be solved is not (1) but rather a perturbed system to unfold the singularity; that is, F(y, λ) + ηr = 0; η  R and , r  Range(Fy, Fλ) and η will be identically zero at the bifurcation point. Also, to guarantee the unique solvability of the extended system, a similar condition like the one in (6) must hold and it will be a symmetric 2 × 2 matrix, see Attili [1], [2].

In the cubic turning point case, the problem will be a two parameter one; that is,F(y,λ,μ)=0.To compute such singular points will require the extension twice to produceF(y,λ,μ)=0,g(y,λ,μ)=0,g˜(y,λ,μ)=0,since the cubic turning point (y0, λ0, μ0) for F(y, λ, μ) = 0 with respect to one parameter corresponds to a simple turning point for (4) with respect to the other one. More details on these ideas can be found in Attili [1], [4] and Spence and Werner [12].

The outline of the remaining sections of this paper will be as follows. In Section 2, we will present the discretization of the B.V.P’s through projection methods, together with some convergence results. Finally, in Section 3 we will consider the details of the collocation and numerical examples.

Section snippets

Discretization using projection methods

Let us consider the application of the results obtained in the previous section to a simple problem which is sufficient to illustrate the results,y+f(y,λ)=0,axb;y(a)=y(b)=0.To find v and g the problem to be solved is of the formLv+gr=v+f(y,λ)v+gr=0,v(a)=v(b)=1TTv=v(1/2)=1,where L:C2[a, b]  B  C[a, b] and B = {y:y(a) = y(b) = 0}. While to find u and g, we solveLu+gT=0,ur=1,where L is the adjoint of L. Now, to solve (9), (10), (11) numerically, we must deal with the finite dimensional forms of

Collocation at gauss points and numerical examples

In this section, we will describe the collocation at Gaussian points. For that reason let us work with a more general two point boundary value problem of the formL(u)=u(m)(s)+k=0m-1ek(s,λ)u(k)(s)=f(s,λ);asb;λR,subject to the m– linearly independent homogenous boundary conditionsk=0m-1[αiku(k)(a)+βiku(k)(b)]=0;1im,where αik and βik are constants.

To describe our collocation scheme we start by defining Πn:a = s0 < s1 <⋯< sn = b to be a partition of [a, b], Pn, k, p): the family of Cp[a, b] functions

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