Numerical quadrature and two-dimensional radiative heat transfer

The paper discusses four methods for the numerical integration of a two-variable integral equation for the radiative flux in a diffuse grey cavity represented by a semi-infinite congruent parallel strip geometry. Method 1 involves only the application of Gauss-Legendre quadrature to the nondimensional finite x-segment and Gauss-Laguerre quadrature to the nondimensional infinite y-segment. In Method 2, the infinite segment is inverted onto a finite segment, and Gauss-Legendre quadrature is used in each direction. Method 3 is similar to Method 2 except that the infinite segment is divided into at least one finite segment and an infinite segment that is inverted. Method 4 is similar to Method 3 except that physical reasoning rather than inversion is used, thus eliminating a numerical integration in the finite segment. It is shown that Method 4 is the only technique which can be used to bring about a convergence to an acceptable solution as the order of quadrature increases. The flexibility of Method 4 is discussed for different parameters.