A Conservative Uniformly Accurate Difference Method ...

SIAM J. NUMER. ANAL.

(C) 1986 Society for Industrial and Applied Mathematics

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Vol. 23, No. 6, December 1986

010

A CONSERVATIVE UNIFORMLY ACCURATE DIFFERENCE METHOD FOR A SINGULAR PERTURBATION PROBLEM IN CONSERVATION FORM* ALAN E. BERGERf Abstract. A conservative three point finite difference method is presented for the numerical solution of the singular perturbation problem eUxx + (b(x)u), =f(x), 0 < x < 1, u(0) and u(1) given, b(x) O. Certain a priori estimates are established for this problem and are then used to obtain uniform error estimates for the difference scheme. Some illustrative numerical results are also given.

Key words, conservative difference method, singular perturbation problem, uniformly accurate AMS(MOS) subject classifications. Primary 65L10, 65M05, 65M10; secondary 34E15

1. Introduction. This paper is devoted to the analysis of a "conservative" three point difference method for the numerical solution of the singular perturbation problem in conservation form Lu=- eUxx+(b(x)u),=f(x) forO B1 on [0, 1] for some positive constant B1. The latter assumption disallows the presence of turning points; also, one can observe that the case of b(x) < 0 can be transformed to b(x) > 0 with the change of variable ;/= 1 x. No restriction on the sign of bx(x) will be imposed. Under these assumptions, problem (1.1) will be seen in 3 to satisfy the same type of a priori estimates as obtained by Kellogg and Tsan 12] for the nonconservation form equation--i.e., the term (b(x)u), in (1.1) being replaced by b(x)ux(x). A conservative three-point difference scheme for (1.1) (in the sense defined in (2.7) below), which is a modification of the "scheme (C)" developed by Kellogg, Shubin and Stephens [11] for Burgers’ equation, will be described in 2, and its rate of convergence as a function of e and the mesh size will be stated and discussed. The scheme is well defined for arbitrary b(x); however, the error analysis to be presented here depends on the restriction b(x) O. In 4 the a priori estimates on the solution of (1.1) will be used in a comparison function analysis, like that in [12], [3], [4], to obtain the rate of convergence of the difference method for (1.1). Finally, some illustrative numerical results will be given. Niijima [18] has previously derived and used this difference replacement for Lu= eux,+(b(x)u)x to define and analyze a difference method for the semilinear problem

(1.2)

Lu-d(x,u)=O for0O,b>O,d>-O.

* Received by the editors July 3, 1984, and in revised form October 15, 1985. This work was supported jointly by the Office of Naval Research under contract N001484WR24012, contract ID NR044-517, and by the Naval Surface Weapons Center Independent Research Fund. Applied Mathematics Branch-R44, Naval Surface Weapons Center, Silver Spring, Maryland 20910. 1241


1242

ALAN E. BERGER

Their scheme is discussed further in 2.1. Difference methods for problems involving the fully nonlinear conservation form operator eUxx-(g(U))x-C(X, u) are examined in, e.g., [2], [13], [17], [19], [23].

2. The difference scheme. Let J be a positive integer and define the uniform mesh 1/J. Let the grid points {x} be given by =jh, j =0, 1,..., J, denote by the value (to be determined) for u u(x), and set p b(xj)h/e. The approximate U scheme to be considered will be written in the form

length h

--

eh-:(rf Uj_ gf Uj F; Uj+I)

(2.1a) for j

x

1,

, J- 1, with

qff-i + qf + qff+l

(2.1b) Uo ao and Here denotes f(x) etc., and the choice of the coefficients rf,

r,

rf and q-f, q.j, qf determines the particular three point scheme. When it will be clear from the context, the j subscripts in rf,..., qf will be omitted. The following shortened notation will also be employed: given an arbitrary set of values { V}, define

_, r; V r; _, q; v

RV =- eh-2(rf

(2.2a) and

Q v qf

(2.2b)

+

+ Vj+I)

+ qf +1.

+

Then (2.1 a) can be written as R U Qf The scheme to be studied is given by

r (2.3

,.+

e-"-’ 1 e-’

tOj_l

Pj+I

r)

-p(1 + e-") 1

e-’

qf (1- rf )/ (2p_,),

q+. =(rf-1)/(2p+l)

qC. _! -2.

This scheme is quite similar to the E1-Mistikawy-Werle scheme [6] which is uniformly second order accurate for the nonconservation form of (1.1) [3], [8], [21], [30], [32], cf. also [33]. The coefficients defined in (2.3) only depend on {bj_l, bg, b+l, h, e}; thus no interpolation of u between grid points would be necessary to formally apply this scheme to the case in which b or f depends on u. Using the difference operators

D+D_(V)= h-2(V_I-2V+ V+I), Do()=(+,- V-l)/(2h), the quantity RgV given by (2.3), (2.2a) can also be written as

where coth x by

(2.5)

(e + e-X) (e

e-X). In 11 a corresponding operator R was defined

RV= eD+D_([p coth p] V) + Do(bV).

Since f was zero for the problem studied in 11 ], no_Q operator was specified there. For fixed e > 0, the two difference operators R and R both "approach" the standard centered scheme as h 0 [29]. For the model problem (1.1) it can be seen that as e 0


CONSERVATIVE DIFFERENCE SCHEME

1243

for h fixed, the difference method RjU=Qjf approaches (bj+lU+l-bjU)/h= (f +f+1)/2 while the quantity RV approaches (b_l V_I 4bV + 3b+1V+)/(2h). 2.1. Rate of convergence. For any function g(x) and nonnegative integer m, the norm of g in C’[0, 1], i.e., the maximum magnitude of g and its first m derivatives on [0, 1], will be denoted by Iglm" In 4 we will prove THEOREM 2.1. There is a positive constant C depending only on {ao, a, Ifl, Ib14, B such that if u is the solution of 1.1 ), { U } is the solution of (2.1), (2.3), and e is in (0, 1 ], then for j O, 1,. J

,

when h -O is in cm-l[0, 1], and f-f(x, e) is such that for i-O,..., m-l, D,f(x, e) exists and is continuous in 0