A class of quasilinear, singularly-perturbed, two-point, boundary value prob- .... where u~ +1 for each fixed m is the solution of the following terminal value.
NORTH- HOUAND
A Computational Method for Solving Quasilinear Singular Perturbation Problems J. Jayakumar and N. R a m a n u j a m
Department of Mathematics Bharathidasan University TiruchirapaUi-620 023 India
Transmitted by John Casti
ABSTRACT A class of quasilinear, singularly-perturbed, two-point, boundary value problems for second-order, ordinary differential equations without interior turning points is considered. To solve these problems Newton's method of quasilinearization is adopted. Then the resultant linear problems are solved by the numerical method suggested in [1]. The method presented in [1] is a combination of an exponentially-fitted finite difference method and a classical numerical method. Further, it is based on the boundary value technique [2] generally used to solve singularly-perturbed boundary value problems. Error estimates for the numerical solution of linear problems are stated. Some examples are given to illustrate the method.
1.
INTRODUCTION
Numerical treatment of singular perturbation problems has received a great deal of attention in the past few decades. Moreover, these problems arise very frequently in many branches of mathematics such as fluid mechanics, elasticity, chemical reactor theory, and in many other allied areas. It is well known that the solutions of singularly-perturbed boundary value problems (SPBVPs) have thin transition layer(s) where the solution behaves very rapidly, while away from the layer(s) the solution behaves
APPLIED MATHEMATICS AND COMPUTATION 71:1-14 (1995) (~) Elsevier Science Inc., 1995 655 Avenue of the Americas, New York, NY 10010
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J. JAYAKUMARAND N. RAMANUJAM
regularly and varies slowly. Hence, numerical treatment of these problems poses major computational difficulties. In this paper, we consider the following SPBVPs of the form
syxx=f(x,y, yx)=-a(x,y)y~+b(x,y), y(O)=A,
x e~,
y(1)=B,
(la) (lb)
where ~ is a small positive parameter and f2 := [0, 1]. We assume that a(x, y) and b(x, y) are sufficiently smooth functions satisfying the following conditions:
la(x,y)l > c > O,
b(x,y) > d > O,
x E ~,
y E R.
(2)
By these assumptions, (1) has a unique solution y(x) = y(x, ~) and has a boundary layer of width O(E) near x = 0 or x = 1 [3]. From the literature of numerical methods to solve (1), we find Pearson [4] was perhaps the first one who tried to solve SPBVPs of the form (1) numerically by taking net adjustments in finite difference methods. Abrahamson et al. [5] suggested what is called directional difference methods for a general class of linear SPBVPs which yielded accurate solution in the outer region (that is, away from the boundary layer). Flaherty et al. [6] have developed an algorithm to solve (1) numerically. In fact, they considered a general class of SPBVPs and used singular perturbation theory to construct formal asymptotic expansion of the solution and solved for these leading terms using standard numerical methods. Lorenz [7] proposed a combination of initial and boundary value technique for solving (1) numerically. Doolen et al. [8] solved (1) by linearizing the nonlinear equation and then solved the resultant linear problems using various exponentially fitted difference schemes. Gasparo et al. [9, 10] presented an initial value technique for solving (1) numerically. Other works in this area include Bender [11], and O'Malley [3]. The numerical method suggested in this paper for the quasilinear SPBVP (1) is as follows. We use Newton's quasilinearization procedure to obtain a sequence of linear SPBVPs. Then using the boundary value technique suggested in [1] the linear SPBVPs are solved. That is, we divide the domain ~ of the differential equation into inner and outer solution regions and solve the differential equations on these regions separately. The terminal boundary condition for the inner region problem is obtained using the zero-order asymptotic expansion of the solution. Further, we use discrete invariant embedding algorithm [12] to solve the system of linear algebraic equations arising from the discretization of the differential equations. Error estimates for the numerical solution of linear problems are stated.
Singular Perturbation Problems
3
Section 2 describes the present computational method in detail. A few examples are given in Section 3 to illustrate the method.
2.
D E S C R I P T I O N OF T H E M E T H O D
To solve (1), we use the linearization process and obtain the Newton sequence { y m } ~ for the initial guess y0 satisfying the boundary conditions: y°(0) -- A,y°(1) = B. Thus we define ym+l, for each fixed m, to be the solution of the following linear SPBVPs:
ey m+l + am(x)yxm+l - bm(x)y m+l = fro(X), ym+l(0) ----A,
X • ~,
ym+l(1) = B,
(3a) (3b)
where m ~ 0 and a m (x), bm (x), and f m (x) are given by
am(x) = a(x, ym), bm(x) = - a " ,ym"m)yx + bv(x, ym), f m ( ~ ) = b(~, v m) - bm(~)V m. If the initial estimate y°(x) is sufficiently close to y(x), then the Newton sequence {ym}~ converges to y(x). From (2), it follows that
laml ~ la(x,y'~)l ~ c > o,
x • ~,
ym • R.
(4)
In the following we consider the case a(x, ym) _> c > 0. The other case a(x, ym) < c < 0 can be put into the form of first case by the change of the independent variable from x to 1 - x. Further, we assume that
bin(x) = _au(x, ym)y~ + by(x, ym) >_ d >_ O,
x e 12,
ym • R. (5)
REMARK 1. The condition (5) should be checked before applying the present computational method. The differential equations in Examples 1-2 satisfy this condition after linearization. Hence, for each fixed m, we solve (3) using the computational method given in [1]. In the following we describe this method briefly. Let k > 0 be a constant such that k¢ 0, we have lyre+l(2, ~) - ~ m + l (2)[ < C~,
x • ~,
where y,~+l is the solution of (3) and ~m+l is given by (7).
2.2. Inner region problem The inner region problem for (3) for each fixed m is given by C .
m+l m m+l ~ + ~ (~)y~ - b ~ ( ~ ) y ~+~ = f ~ ( ~ ) ,
ym+l(0) = A,
Ym+l(ke) = Am+l.
• e ~1,
(9a)
(9b)
Singular Perturbation Problems
5
To solve (9) numerically, we use the exponentially fitted difference scheme [8] for each fixed m: e a ( P ) D + D - Y ~ +1 + a m l~x i)~Doyim + l
- bm (xi)y im+l = f m ( x i ) ,
"
i=l(1)Y-1,
y~n+l : A,
y ~ + l = Am+l,
(10a) (10b)
where xi = ih,
h = 1iN,
D - y i = (Yi - y i - 1 ) / h ,
D+yi = (Yi+I - y i ) / h ,
i = 0(1)N,
Doyi = (Yi+l - y i - 1 ) / 2 h ,
and a(p) = p a m ( O ) c o t h ( p a m ( O ) / 2 ) / 2 , 2.3.
p=h/e
Outer region problem
The outer region problem for (3) for each fixed m is given by X e ~2,
~ymx'kl "]- a m ( x ) y r ~ +1 -- b m ( x ) y m+l = f r o ( x ) ,
Ym+X(k¢) : Am+l,
ym+l(1) = B.
(lla) (lib)
To solve (11) numerically for each fixed m we use the following upwind scheme: e D + D - Y m+l + amlx~ ij~D+yi"m+l _ bm(x i)Yi ~. m+l = f m ( x ~ ) '
i=l(1)N-1,
y~n+l ---- 2~m+l,
y ~ + l = S.
(12a) (12b)
REMARK 2. For each fixed m, systems (10) and (12) are tridiagonal which are solved by the method of discrete invariant embedding [12]. REMARK 3. Here we have denoted the mesh size with the same symbol h, although its value is different in inner and outer solution regions. 2.4.
Error estimate
The following theorems give error estimates for the inner and outer region problems, respectively. Since the proofs of these theorems are analogous to the Theorem 2 and 4 stated in [1], we state here only the theorems and omit the proof.
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J. JAYAKUMAR AND N. RAMANUJAM
THEOREM 1. Let ymTl(x) and y~+l be the solutions of (3) and (10), respectively. Then, for each fixed m, we have
lyrn+l(xi) -
y~n+lI _~ C(h + 6),
xi e ~1,
i = 0(1)Y,
where C is a constant independent of i, h, 6. THEOREM 2. Let ym+l(x) and ym+l be the solutions 0](3) and (12), respectively. Then, for each fixed m, we have
lym+l(xi) - y y + l I < C[h + 6 + hexp(-Txi/6)/6],
lym+l(xi)--ym+l[
h
