A fast iterative method to compute the flow around a submerged body

Abstract

We develop an efficient iterative method for computing the steady linearized potential flow around a submerged body moving in a liquid of finite constant depth. In this paper we restrict the presentation to the two-dimensional problem, but the method is readily generalizable to the three-dimensional case, i.e., the flow in a canal. The problem is indefinite, which makes the convergence of most iterative methods unstable. To circumvent this difficulty, we decompose the problem into two more easily solvable subproblems and form a Schwarz–type iteration to solve the original problem. The first subproblem is definite and can therefore be solved by standard iterative methods. The second subproblem is indefinite but has no body. It is therefore easily and efficiently solvable by separation of variables. We prove that the iteration converges for sufficiently small Froude numbers. In addition, we present numerical results for a second-order accurate discretization of the problem. We demonstrate that the iterative method converges rapidly, and that the convergence rate improves when the Froude number decreases. We also verify numerically that the convergence rate is essentially independent of the grid size.