An index reduction method for linear Hessenberg systems

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Abstract

In [E. Babolian, M.M. Hosseini, Reducing index, and pseudospectral methods for differential–algebraic equations, Appl. Math. Comput. 140 (2003) 77–90] a reducing index method has been proposed for some cases of semi-explicit DAEs (differential algebraic equations). In this paper, this method is generalized to more cases. Also, it is focused on Hessenberg index 2 systems and proposed reduction index method will be illustrated for this problem. The Hessenberg system and its obtained reduced index system are numerically solved through pseudospectral method. In addition, aforementioned methods will be considered by one example.

Introduction

Consider a linear (or linearized) semi-explicit DAEsX(m)=j=1mAjX(j-1)+By+q,0=CX+r,where Aj and C are smooth functions t, t0  t  tf, Aj(t)  Rn×n, j = 1,  , m, B(t)  Rn×k, C(t)  Rk×n, n > k, and CB is nonsingular (the DAEs has index m + 1). The homogeneities are q(t)  Rn and r(t)  R. It is well-known that the DAEs (1a), (1b) can be difficult to solve when it has a higher index (index greater than one, [1]). In this case an alternative treatment is the use of index reduction methods (see, e.g., [4], [5], [7], [9]), until a well-posed problem (index-1 DAEs or ordinary differential equations) is obtained. Early in [3] and [8], for k = 1, the index of problem (1a), (1b) has been reduced by introducing a simple formulation. In this paper, we will reduce the index of (1a), (1b) when k > 1. For this reason, we puty=(CB)-1CX(m)-j=1mAjX(j-1)-q,and by substituting (2) in (1a), we obtain an implicit DAE which has index m, as follows:j=0mEjX(j)=qˆ,where Ej(t)  Rn×n, j = 0,1,  , m, and except E0(t), others are singular matrices. Note that system (3) has k equations less than system (1a), (1b).

It is known that the eigenfunctions of certain singular Sturm–Liouville problems allow the approximation of functions C[a, b] where truncation error approaches zero faster than any negative power of the number of basic functions used in the approximation, as that number (order of truncation N) tends to infinity [6]. This phenomenon is usually referred to as “spectral accuracy” [6]. The accuracy of derivatives obtained by direct, term-by-term differentiation of such truncated expansion naturally deteriorates [2], but for low-order derivatives and sufficiently high order truncations this deterioration is negligible, compared to the restrictions in accuracy introduced by typical difference approximations (for more details, refer to [3], [8]). Throughout, we are using first kind orthogonal Chebyshev polynomials {Tk}k=0+ which are eigenfunctions of singular Sturm–Liouville problem1-x2T(x)+k21-x2Tk(x)=0.

Section snippets

A simple formulation for index reduction

In this section, DAEs (1a), (1b) is considered when m = 1 and k = 2. To extend it to general case (1a), (1b) is easy. Now consider the Hessenberg index-2 system,X=AX+By+q,0=CX+r,where A = (aij)n×n, B = (bij)n×2, C = (cij)n, n  3 anddet(CB(t))0,t[t0,tf].From (4a) and (5), we can writey=(CB)-1C[X-AX-q],t[t0,tf],and substituting (6) into (4a), implies,X=AX+B(CB)-1C[X-AX-q]+q.So, problem (4a), (4b) transforms to the systemdet(CB(t))[I-B(CB)-1C][X-AX-q]=0,CX+r=0.Here, the overdetermined system (7a),

Implementation of numerical method

Here, the implementation of pseudospectral method is presented for DAEs systems (4a), (4b) and (8). This discussion can simply be extended to general forms and (3). Now consider the DAEs systems,j=13fj(t)xj+j=46fj(t)xj-3=qˆ(t),j=13cij(t)xj=-ri(t),i=1,2,with initial condition,x1(t0)=α.For an arbitrary natural number ν, we suppose that the approximate solution of DAEs systems (14a), (14b) is as below,xj(t)=i=0νai+(j-1)×(ν+1)Ti(s),j=1,2,3,s[-1,1],wheret=h(s)=tf-t02s+tf+t02,where a = (a0, a1,  , a3

Numerical example

Here, we use “ex” and “ey” to denote the maximum absolute error in vectors X = (x1, x2, x3) and y = (y1, y2). These values are approximately obtained through their graphs. Results show the advantages of techniques, mentioned in Sections 2 A simple formulation for index reduction, 3 Implementation of numerical method. Also, the presented algorithm in Section 3, is performed by using Maple 8 with 25 digits precision.

Example

Consider for −1  t  1,X=AX+By+q,0=CX+r,whereA=3-2t2-t0012-t-1000-1,B=4-2t001sin(2t)cos(2t)

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    Hence stabilized index reduction methods were used to overcome the difficulty. Earlier in [3,5–7], for semi-explicit DAEs, an efficient reducing index method has been proposed that does not need the repeated differentiation of the constraint equations. This method is well applied to DAEs with and without singularities and then are numerically solved by pseudospectral method with and without domain decomposition.

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