On the numerical evaluation of a particular singular two-dimensional integral

Authors:
G. Monegato and J. N. Lyness

Journal:
Math. Comp. **33** (1979), 993-1002

MSC:
Primary 65D30; Secondary 65B05

DOI:
https://doi.org/10.1090/S0025-5718-1979-0528052-6

MathSciNet review:
528052

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Abstract: We investigate the possibility of using two-dimensional Romberg integration to approximate integrals, over the square , , of integrand functions of the form where 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 has a point singularity of a specified nature at the origin.

**[1]**R. L. BISPLINGHOFF, H. ASHLEY & R. L. HALFMAN,*Aeroelasticity*, Addison-Wesley, Reading, Mass., 1957, pp. 188-293.**[2]**J. N. LYNESS, "Symmetric integration rules for hypercubes. III. Construction of integration rules using null rules,"*Math. Comp.*, v. 19, 1965, pp. 638-643. MR**0201069 (34:954)****[3]**J. N. LYNESS, "An error functional expansion for*N*-dimensional quadrature with an integrand function singular at a point,"*Math. Comp.*, v. 30, 1976, pp. 1-23. MR**0408211 (53:11976)****[4]**J. N. LYNESS & B. W. NINHAM, "Asymptotic expansions and numerical quadrature,"*Math. Comp.*, v. 21, 1967, pp. 162-178. MR**0225488 (37:1081)****[5]**C. S. SONG, "Numerical integration of a double integral with a Cauchy-type singularity,"*AIAA J.*, v. 7, 1969, pp. 1389-1390. MR**0245204 (39:6516)****[6]**W. SQUIRE, "Numerical evaluation of a class of singular double integrals by symmetric pairing,"*Internat. J. Numer. Math. Engrg.*, v. 10, 1976, pp. 703-708. MR**0455307 (56:13546)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1979-0528052-6

Article copyright:
© Copyright 1979
American Mathematical Society