Efficient Implementation of the Exact NumericalFar Field Boundary Condition for PoissonEquation on an Infinite Domain

Computation of exterior fluid problems often involves solving the Poisson equation in an unbounded domain. To cut down computational cost, it is desirable to minimize the computational domain. Therefore an important numerical issue is how to specify correctly the boundary condition at a finite computational boundary. In this note, we present an efficient method for such a task. The basic idea behind the current work and many previous works (see, for example, [1–4]) exploits solutions to the Laplace equation outside the compact support of the source. Therefore, in theory, all these methods are equivalent. However, the current numerical implementation is simpler than that of previous methods. In particular, we solve the Poisson equation with a false Dirichlet boundary condition, from which the correct solution can be obtained. This avoids handling mixed boundary conditions in the far field such as those used by Keller and Givolli [1]. Moreover, our strategy is independent of the discretization scheme; thus the derivation is more transparent than derivations employing explicit differencing schemes, as in the work of Anderson and Reider [3]. Because our method is independent of the discretization scheme, it can be implemented straightforwardly using a variety of discretization schemes, including finite difference and finite element methods. The current method can also be more efficient in high-order schemes than other methods involving inverting matrices whose complexity depends on the order of finite differencing [4]. Finally,