The nonlinear singular integral equation which is of interest to researchers in transonic flow is critically examined, and reasons are advanced to indicate why standard numerical techniques are not satisfactory in solving this type of equation. The inherent difficulties in the approximation of the integral term of this equation are pointed out, and special techniques are proposed to mitigate the inaccuracies resulting from the standard approximations. The basic principle for each of these techniques is the same, but the orders of approximation differ; for each such approach, numerical results are given together with the computational time involved. Wherever possible, estimates for the error have been included; it is also shown how the integral may be bounded from above and below. Though the essence of the paper is numerical, it also contains certain analytic features; it has been shown, for example, how the infinite domain of integration may be reduced to a finite one, and how the singular kernel may be reduced to a non-singular one. Numerical results have been plotted and compared with those of earlier investigators and it is seen that the methods described here are an improvement over the preceding works, either in numerical accuracy or computational economy.

The nonlinear singular integral equation of transonic flow is examined in the free-stream Mach number range where only solutions with shocks are known to exist. It is shown that, by the addition of an artificial viscosity term to the integral equation, even the direct iterative scheme, with the linear solution as the initial iterate, leads to convergence. Detailed tables indicating how the solution varies with changes in the parameters of the artificial viscosity term are also given. In the best cases (when the artificial viscosity is smallest), the solutions compare well with known results, their characteristic feature being the representation of the shock by steep gradients rather than by abrupt discontinuities. However, ‘sharp-shock solutions’ have also been obtained by the implementation of a quadratic iterative scheme with the ‘artificial viscosity solution’ as the initial iterate; the converged solution with a sharp shock is obtained with only a few more iterates. Finally, a review is given of various shock-capturing and shock-fitting schemes for the transonic flow equations in general, and for the transonic integral equation in particular, frequent comparisons being made with the approach of this paper.