On the Numerical Evaluation of a Particular Singular Two - Dimensional Integral

We investigate the possibility of using two-dimensional Romberg integration to approximate integrals, over the square 0 < x, y < 1, of integrand functions of the form g(x, y)l(x y) where g(x, y) is, for example, analytic in x and y. We show that Romberg integration may be properly justified so long as it is based on a diagonally symmetric rule and function values on the singular diagonal, if required, are defined in a particular way. We also investigate the consequences of ignoring fhese function values (i.e. setting them to zero) in the context of such a calculation. We also derive the asymptotic expansion on which extrapolation methods can be based when g(x, y) has a point singularity of a specified nature at the origin. 1. The calculation of the aerodynamic load on a lifting body occasionally requires the calculation of a two-dimensional Cauchy principal value integral of the type •i-K*. y) Jo Jo dx dy. y (See, for example, Bisplinghoff, Ashley and Halfman [1].) One approach to the evaluation of such integrals has been described by Song [5]. In this paper we show that, with some minor provisos, a straightforward application of two-dimensional Romberg integration may be used in this problem. The two-dimensional integral (1.1) If = ¡l¡]Qf(x,y)dxdy may be approximated numerically using a two-dimensional quadrature rule Q defined by (1.2) Qf= ¿ wfiXj.yj), ¿W/ = -. 7=i /=This rule has polynomial degree d(Q) when (1.3) Qf=If for all/G nd, where rrd is the set of polynomials of degree d or less. Definition 1.4. A symmetric rule Q is one for which (1.4) Qf = Qf for all 7(1 x, 1 y) = f(x, y). Received April 17, 1978. AMS (MOS) subject classifications (1970). Primary 65D30, 47A55; Secondary 65B05, 65B15. *Work performed under the auspices of the Italian Research Council. **Work performed under the auspices of the U.S. Department of Energy. © 1979 American Mathematical Society 0025-5718/79/0000-0107/S03.50 993 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 994 G. MONEGATO AND J. N. LYNESS The m -copy of the rule Q is defined by m-l m-\ v W; ¡X; + k, V; + /C,\ (1.5) ß(«)/= x x E-,/ J—L>-—-]■ fc.=o k2 = o /=i m1 \ m m J This is a weighted sum of function values obtained by subdividing the unit square into m2 equal squares of side \/m and applying a properly scaled version of Q to each. Q}1"^ has the same polynomial degree d(Q) and is symmetric when Q is symmetric. Theorem 1.6. Let Qf and Q^m^f be defined as above. Let f(x, y) be a function, all of whose derivatives f-r' s'(x, y) whose total order satisfies r + s < p are integrable over the unit square. Then i-\ B,(Q;f) (1.6) !2(m)/-//=Z -J—r+ <Km-'), 2<KP, i=i ms