Artificial Boundary Method

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.

In this chapter, we discuss the global ABCs for the exterior problem of 2-D and 3-D Poisson equation, the modified Helmholtz equation, and the Helmholtz equation. By using artificial boundaries, the original problems are reduced to boundary value problems on bounded computational domains. Boundary conditions on the artificial boundaries are obtained, and then the finite element method is applied to solve the reduced problems. Some error estimates are also given.

In this chapter, we discuss the global ABCs for the exterior problem of 2-D and 3-D Navier system, and 2-D Stokes system. By using artificial boundaries, the original problems are reduced to boundary value problems on bounded computational domains. Boundary conditions on the artificial boundaries are obtained, and then the finite element method is applied to solve the reduced problems. Some error estimates are also given.

In this chapter, we discuss the global ABCs for heat and Schrödinger equations on unbounded domains. By using artificial boundaries, the original problems are reduced to initial boundary value problems on bounded computational domains. Boundary conditions on the artificial boundaries are obtained, and then the finite difference method is applied to solve the reduced problems. Stability and error estimates are also discussed.

In this chapter, we discuss the ABCs for wave equations, Klein- Gordon equations, and linear KDV equation on unbounded domains. By using artificial boundaries, the original problems are reduced to initial boundary value problems on bounded computational domains. Absorbing boundary conditions on the artificial boundaries are obtained, and then the finite difference method is applied to solve the reduced problems. Some stability results are also given.

In this chapter, we discuss the local ABCs for the exterior problem of 2-D and 3-D Poisson equations, and for the wave equations on unbounded domains. By using artificial boundaries, the original problems are reduced to boundary or initial boundary value problems on bounded computational domains. Local boundary conditions on the artificial boundaries are obtained. Some error estimates are also given.

In this chapter, we discuss the discrete ABCs for the exterior problem of 2-D Poisson equation, 2-D Navier-Stokes equations, and 2-D linear elastic system, and for the 1-D Klein-Gordon equation on unbounded domains. By using artificial boundaries, the original problems are reduced to boundary or initial boundary value problems on bounded computational domains. Discrete boundary conditions on the artificial boundaries are obtained, and then the finite element or finite difference method is applied to solve the reduced problems.

In this chapter, we discuss the implicit ABCs for the exterior problem of 2-D and 3-D Poisson equations, the Helmholtz equation, and the Navier system, and for the wave equation on unbounded domains. By using artificial boundaries, the original problems are reduced to boundary or initial boundary value problems on bounded computational domains. Implicit boundary conditions on the artificial boundaries are obtained, and then the finite element or finite difference method is applied to solve the reduced problems. Some error estimates are also given.

In this chapter, we discuss the nonlinear ABCs for Burgers equation, Kardar-Parisi-Zhang equation, and Schrödinger equation on unbounded domains. By using artificial boundaries, the original problems are reduced to initial boundary value problems on bounded computational domains. Nonlinear boundary conditions on the artificial boundaries are obtained, and then the finite difference method is applied to solve the reduced problems. Some stability results are also given.

In this chapter, we discuss the application of ABCs for some problems with singularity, including the modified Helmholtz equation with singularity, the interface problem, the linear elastic system with singularity, and the Stokes equations with singularity. By using artificial boundaries, the singular points are removed, and the original problems are reduced to boundary value problems on computational domains. Boundary conditions on the artificial boundaries are obtained, and then the finite element method is applied to solve the reduced problems. Some error estimates are also given.