Proceedings of the Third International Colloquium on Numerical Analysis: Plovdiv, Bulgaria, 13–17 August, 1994 [Reprint 2020 ed.] 9783112314098, 9783112302828

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Proceedings of the Third International Colloquium on Numerical Analysis

Proceedings of the

Third International Colloquium on Numerical Analysis Plovdiv, Bulgaria, 1 3 - 1 7 August, 1994

Editors: D. Bainov and V. Covachev

HI \ISPIII

Utrecht, The Netherlands, 1995

VSP BV P.O. B o x 3 4 6 3 7 0 0 A H Zeist The Netherlands

© V S P B V 1995 First p u b l i s h e d in 1 9 9 5 ISBN 90-6764-193-6

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

Printed in The Netherlands

by Koninklijke

Wöhrmann bv,

Zutphen.

CONTENTS Preface

1

Difference Methods for First Order Partial Differential Equations with Impulses D. Bainov, Z. Kamont and E. Minchev

3

Numerical Solutions of Singular Nonlinear Boundary Value Problems J. V. Baxley

15

A Survey: Continuous Selections for Metric Projections A.L. Brown

25

Solutions of Btitzer's Problems (Linear Form) and Some Algebraic Polynomial Operators with Saturation Order 0(n 2) J.-D. Cao andH.H. Gonska

37

Advanced Steppoint Methods for Initial Value Problems J R. Cash and Y. Psihoyios

43

Construction of 3-D Meshes Using the Triangulation of Delaunay J. M. Escobar and R. Montenegro

51

Convergence of a Finite Volume Scheme for a Nonlinear Hyperbolic Equation R. Eymard, T. Gallouet and R. Herbin

61

Overview of QR Methods for Large Least Squares Problems Involving Kronecker Products D. W. Fausett and H. Hashish

71

A Conservative B-Spline Finite Element Method for the Nonlinear Schroedinger Equation L.R.T. Gardner and G.A. Gardner

81

Global Smoothness Preservation by Blossoms of Bernstein Polynomials H.H. Gonska, H.-J. Wenz andD.-X. Zhou

91

A Four Point Finite Volume Scheme for a Diffusion Convection Problem on a Triangular Mesh R. Herbin and O. Labergerie

97

Surfaces Associated to Real Zeros ofHolomorphic Functions with Parameters J. Kajiwara and M. Tsuji

107

vi

contents

Monotone Iterations for Numerical Solutions of Nonlinear Convection-Diffusion Equations X. Liu

117

Recursive Triangles G. Muhlbach

123

An Approach to Automatic Finite Element Mesh Generation by Managing Binary Images A. Plaza, R. Montenegro and L. Ferragut

135

A Matrix Formulation of Automatic Differentiation L.B. Rail

143

Numerical Aspects of Multivariate Interpolation and Approximation M. Reimer

151

S-Convexity Preserving Interpolation of Gridded Three Dimensional Data Using Rational C1 Splines J.W.Schmidt

161

Essentially Non-Oscillatory Finite Difference, Finite Volume and Discontinuous Galerkin Finite Element Methods for Conservation Laws C.-W. Shu

171

An Algorithm for Sum of Floating-Point Numbers without Round-Off Error K. Tsuji

181

A Multi-Peak System Identification M. Tsuji

191

A Blossoming Approach to Quasiinterpolants H.-J. Wenz Parallel Algorithms for Linear Algebra on a Shared Memory Multiprocessor K. Wright and D. Kaya

201

209

Parallel Implementation of a Class of Nonstationary Alternating-Type Methods D M. Young and D R. Kincaid

219

Time Parallel Methods for Ordinary Differential Equations R. Alt

223

Preface The Third International Colloquium on Numerical Analysis was organized by the Mathematical Society of Japan, the Department of Mathematics of the Catania University, and UNESCO with the cooperation of the Association Suisse d'Informatique, the Canadian Mathematical Society, Hamburg University, Kyushu University, the London Mathematical Society, Plovdiv University "Paissii Hilendarski", Technical University — Berlin, Technical University — Plovdiv, Union of the Bulgarian Mathematicians — Plovdiv Section, the University of Texas — Pan American, and sponsored by the foundation "Evrika" and the "Pharmaceutical Scientific and Production Enterprise". It was held on August 13-17, 1994, in Plovdiv, Bulgaria. The subsequent colloquia will take place each year from 13 to 17 of August in Plovdiv, Bulgaria. The address of the Organizing Committee is: Stoyan Zlatev, Mathematical Faculty of the Plovdiv University, Tsar Assen Str. 24, Plovdiv 4 000, Bulgaria. The Editors

3th Int. Coll. on Numerical Analysis, pp. 3-14 D. Bainov and V. Covachev (Eds) © VSP 1995

Difference methods for first order partial differential equations with impulses D . BAINOV1, 1 2

Z. KAMONT2,

E.

MINCHEV1

Higher Medical Institute, Sofia 1504, P.O. Box 45, University of Gdansk, Gdansk, Poland

Bulgaria

A b s t r a c t . The paper deals with a class of difference methods for initial- value problems and for mixed problems for first order partial equations with impulses. We give sufficient conditions for the convergence of a sequence of approximate solutions under the assumptions that the righthand sides satisfy the non-linear esimates of the Perron type with respect to unknown function. The proof of stability is based on monotone techniques. Keywords

1

and phrases:

difference

methods,

impulsive

partial

differential

equations

Introduction

Differential equations with impulses are natural tools for mathematical simulation of many processes in biology, mechanics and population dynamics. T h e papers by V . Milman and A. Myshkis [1] initiated the theory of ordinary differential equations with impulses. At the present moment numerous papers were published concerning various problems for particular classes of equations and special problems such as " b e a t i n g " , dying", "merging", noncontinuability of solutions. Detailed bibliographical information can b e found in [2], [3], [4]. Partial differential equations with impulses were first treated in [5], [6]. Some qualitative properties for parabolic equations and for first order partial equations with impulses were demonstrated in [5], [6j. T h e aim of this paper is to show t h a t there is a class of difference methods for partial equations with impulses. W e formulate the problem. For any two metric spaces X and Y we denote by C(X,Y) the class of all continuous functions from X into Y. Let us denote by E the Haar pyramid E = { ( z , y) e R1+n

: x € [0, o], y = (yi , . . , » „ ) e [ - 6 + Mx,

b -

Mx\)

where a > 0, M = ( M I , . . . , Mn), b = (¡>I,..., f>„), y = (yI,..., yn) andFCJ— MJ a > 0 for i = 1,..., n. Suppose t h a t 0 < a\ < a^ < ••• < o/t are given numbers and aq = 0, afc+i = a. Write E(i) Let Cimp[E,

= {(x,y)

e E : di < x < ai+1},

i = 0,

R\ b e the class of all functions z : E —» R such t h a t

(i) the functions z l^to, i = 0 , 1 , . . . , fc, are continuous,

4

D. Bainov et al.

(ii) for each i, 0 < i < k, (aj, y) G E there exists lim z(t, C) = z(a+,y) (t,C)->(o¿+,y)

(1)

(Hi) for each i, 1 < i < k + 1, (a,, y) € E there exists lim z(tX) (t, CH( -

Mx]}

and E = (E \ Eimp)

x

R

x

£¿ m p =

x ü.

Suppose that / : Y,

R,

g : £¿mp

R,

R

are given functions. We will consider the Cauchy problem with impulses Dxz(x, y) = f (x, y, z(x, y), Dyz(x, y)) for (x, y) 6 E \ Eimp, z(0,y)=lp{y) Az(x,y)

for

= g (x,y,z(x~,y)^j

ye\-b,b}, for

(x, y) e Eimp

(3) (4) (5)

where Dyz = (Dyiz,..., DVnz). We consider classical solutions of problem (3) — (5). A function z : E —> R is a solution of (3) — (5) if (i) 2 € Cimp[E,R] and there exist derivatives Dxz(x,y), Dyz(x,y) for (x,y) 6 E \ Eimp (ii) z satisfies (3) — (5). We are interested in establishing a method of approximating of solutions of the Cauchy problem (3) - (5) by solutions of an associated difference equation and in estimating the difference between the exact and the approximate solution. Difference methods for first order partial equations were first treated by Strang [7], Kowalski [8] and Plis [9]. In this time numerous papers were published concerning various problems for particular cases of hyperbolic equations or systems and for general theory of difference methods. It is not our aim to give a full review of papers concerning the above problems. We shall mention only those which contain such reviews. They are [10], [11], [12] and [13] for differentialfunctional equations. The basic tool in the investigations of the stability of difference schemes are monotone techniques. Theorems on difference inequalities generated by partial equations and recurrent inequalities are used. The monotone methods for nonlinear differential equations based on differential inequalities can be found in [14]. In this paper we prove that there is a natural class of difference methods for first order partial differential equations with impulses. We prove that if the method satisfies a consistency condition and is stable then it is convergent. We give sufficient conditions for the convergence of a sequence of approximate solutions under the assumptions that righthand sides satisfy the non-linear estimates of the Perron type with respect to unknown function. These assumptions are identic with conditions which guarantee the uniqueness of solutions. We will consider initial-value problems and mixed problems with impulses. We use general ideas for finite difference approximations which were introduced in [10], [11], [12], [15]. The proof of stability will be based on monotone technique.

5

Difference methods...

2

Discretization

Let F(X, Y) denote the class of all functions defined on X and taking values in Y where X and Y are arbitrary sets. For y, y € Rn we write y *y = (y\y\,..., ynyn)- Let d = (do, di,..., dn) 6 Rx+n with d, > 0 for i = 0 , 1 , . ,.,n. Suppose that for a h = (ho,h') G (0, d\, h' = (hi,..., hn), there exist No and N = (Ni,..., Nn) where Ni, i = 0, 1,..., n, are natural numbers and Noho = a, N * h' = b. Write |/i| = ho + h\ -f ... + hn We denote by Id the set of all h € (0, d] having the above property. Let Z be the set of integers. For m = (mo, m') € Z1+n, m' = (m1, ...mn) we define x(mo)

= moho,

yW

= m' * h', y™

=

...,y

Suppose that natural numbers are defined by mbers N(01],...,..., N N^^k) a: h0 < ai < (Nq^ + 1) ho, i =

\,...,k.

Put : 0 < m 0 < N0, y(m'} £ [ - è + Mx{mo\

Eh = {(ximo\y^)

E T P = {(^ ( m o ) ,2/ ( m , ) ) : m 0 € {N^,.... y(™') €

b - Mx(mo))}

,

iV0(fc)}

[_;, + M x ( m o ) , b - M i ( m o ) ] } ,

and Eh.o = {î/ (m,) :

-N

< m' < n] .

For z : R we write z(m> = z(x fh

'• Eh.o : —> R-

We will approximate solutions of (3) — (5) by means of solutions of the problem 50z(m)

= fh ( i

w

,j

M

,2

W

,fe

( n l

)

(6)

for(x R+ is the solution of the problem = ,( + h0aH(x®, r/w) + /i R and gh '• £j i mp —> R, h G Id, satisfy the conditions: 1° for each P = (x,y,p,q) G E^, q = (qi,-..,qn), there exist derivatives (DqJh(P),...,DqJh(P))

=

Dqfh(P)

and Dqfh(x, y, p, •) G C{Rn,Rn), 2° /or P G E we have 1 -

n /i0 -j- | DqJh{P) ftj

| > 0, i = l , . . . , n ,

3° there are ah : and ih : X r p

xR

+

^R

+

(12)

Difference

7

methods.

such that \fh(x®ty R is the solution of (6)-(8) and there exists c*o £ such that ^(m') _