Fast direct numerical solution of the nonhomogeneous Cauchy-Riemann equations

Abstract A fast direct (noniterative) “Cauchy-Riemann Solver” is developed for solving the finite-difference equations representing systems of first-order elliptic partial differential equations in the form of the nonhomogeneous Cauchy-Riemann equations. The method is second-order accurate and requires approximately the same computer time as a fast cyclic-reduction Poisson solver (Buneman's method, but with the cyclic reduction of simple tridiagonal matrices replaced by the Thomas algorithm). The accuracy and efficiency of the direct solver are demonstrated in an application to solving an example problem in aerodynamics: subsonic inviscid flow over a biconvex airfoil. The analytical small-perturbation solution contains singularities, which are captured well by the computational technique. The algorithm is expected to be useful in nonlinear subsonic and transonic aerodynamics.