Spurious solutions in a Fredholm integral equation of the second kind with a Cauchy kernel

In this study, we have addressed the appearance of a spurious solution generated when employing low-order accurate schemes for a Fredholm integral equation of the second kind in potential-flow problems, which is highly relevant to moving boundary problems. It is shown that the numerical error resulting from discretization of the Cauchy integral violates the Fredholm solvability condition, which is related to the conservation of the circulation, at the discretization level. In other words, the numerical error is not orthogonal to the homogeneous solution of the adjoint equation and therefore does not comply with the Fredholm compatibility condition. The Fredholm compatibility condition appears to be the total accumulated truncation error requiring circulation preservation at the discretized level. Therefore, in order to obtain a solution, the truncation-error distribution must be redistributed to comply with the requirements imposed by the compatibility constraint. The outcome of violating the solvability constraint is the spurious solution and it seems to be controlled by the numerical diffusion/dispersion terms in the numerical scheme.