On the numerical evaluation of a particular singular two-dimensional integral
G. Monegato and J. N. Lyness
Math. Comp. 33 (1979), 993-1002
Primary 65D30; Secondary 65B05
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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.
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- R. L. BISPLINGHOFF, H. ASHLEY & R. L. HALFMAN, Aeroelasticity, Addison-Wesley, Reading, Mass., 1957, pp. 188-293.
- 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)
- 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)
- J. N. LYNESS & B. W. NINHAM, "Asymptotic expansions and numerical quadrature," Math. Comp., v. 21, 1967, pp. 162-178. MR 0225488 (37:1081)
- 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)
- 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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