Introduction

Many problems in science and engineering are described by partial differential equations on unbounded domains, and must be solved numerically. The flow around an airfoil (see Fig. 0-1), stress analysis of a dam with an infinite foundation (see Fig. 0-2), flow in a long pipe (see Fig. 0-3), and wave propagation in the space (sound wave, elastic wave, electric magnetic wave, etc.) are typical examples. For these problems, the main difficulty is the unboundedness of the domain. Normal numerical methods, such as the finite difference and finite element methods, cannot be applied directly to these problems. One way to solve the problem is to introduce an artificial boundary, and divide the physical domain into two parts: the bounded computational domain and the remaining unbounded domain. The artificial boundary becomes the boundary (or a part of the boundary) of the computational domain. If we can find the boundary condition on the artificial boundary satisfied by the solution of the original problem, then we can reduce the original problem to a boundary value problem on the bounded computational domain, and solve it numerically. In early literature, the boundary condition at infinity is usually applied directly on the artificial boundary. The Dirichlet boundary condition (or Neumann boundary condition) is the commonly used boundary condition. In general, this boundary condition is not the exact boundary condition satisfied by the solution of the original problem, it is only a rough approximation to the exact boundary condition.