Nonlinear Artificial Boundary Conditions

For solving a partial differential equation on an unbounded domain, usually we impose an artificial boundary to truncate the unbounded domain, thus the computed solution is confined to a finite computational domain. With this, and then a suited artificial boundary condition is defined. These problems arise from a lot of areas in science and engineering, such as solid mechanics, fluid mechanics, acoustics, electromagnetics, and quantum mechanics. The construction of the artificial boundary conditions may have a profound influence on the computation of the problem; it is also a very challenging issue in scientific computation. For recent decades, a large number of literatures have been published, which include the design of boundary conditions and numerical schemes, the stability analysis on the induced initial-boundary problems, the error estimation of the numerical schemes, and so on. The so-called artificial boundary method has become one of the most important tools for solving partial differential equations on unbounded domains.In recent years, a lot of new approaches have been introduced in this field for various purposes, such as to rapidly evaluate nonlocal operators of exact artificial boundary conditions, construct higher-order local boundary conditions, solve problems in complex geometries and multi-dimensions, and design artificial boundary conditions for nonlinear problems. In this dissertation we concern with artificial boundary conditions for some nonlinear partial differential equations, which have received much attention from researchers. Due to the nonlinearity of the model equations in the unbounded domain, which invalidates many efficient tools in linear equations such as the Fourier or Laplace transform, we will develop some special techniques to treat the nonlinearity. The dissertation is mainly organized as three parts.First, we investigate absorbing boundary conditions of the nonlinear Schr(o|¨)dinger equations which attract much researchers' interest in recent years. There exist many effective approaches to construct local and nonlocal boundary conditions for the linear Schr(o|¨)dinger equation. However, in the nonlinear case, only a few treatments were proposed, such as the inverse scatting method for the cubic nonlinear Schr(o|¨)dinger equation and the perfectly matched layer method. We develop a local time-splitting method to separate the original problem into a linear equation and a nonlinear equation near the artificial boundary. The method of Fourier transform can then be introduced to the linear part to obtain a one-way equation. For the linear problem, we impose an adaptive strategy through a windowed Fourier transform to capture the wavenumber parameter that is involved in the local absorbing boundary conditions. Furthermore, we apply the so-called split local absorbing boundary method to simulate the laser-atom interactions.Secondly, we investigate exact artificial boundary conditions, which are in nonlocal form, for the nonlinear Burgers equation and the deterministic Kardar-Parisi-Zhang equation. These two nonlinear equations can be related to the linear heat equations through the well-known Cole-Hopf transform. Therefore, some techniques of heat equations on unbounded domains can be adopted. For the Burgers equation, we obtain one dimensional artificial boundary condition, and make a stability analysis of the equivalent initial-boundary value problem confined to the bounded computational domain. In the latter case, we obtain the artificial boundary conditions from one to three dimensions.Thirdly, we discuss an application of the artificial boundary condition in mathematical physics. In such problems as the water waves, it is hoped to obtain a solution of solitary waves. Solving these problems need to consider eigenvalue problems, which are actually attributed to a problem on an unbounded domain. In this part, we give a preliminary study through introduction of some insight, for which the generalized Korteweg-de Vries equations is taken as an example. The analytical tolls are limited for these equations because of the complexity of the nonlinearity. Our approach is to transform the nonlinear eigenvalue problems on the unbounded domain into bounded problems by imposing an artificial boundary condition, where the former problem is deduced from the travelling wave solutions. These numerical solitons are so called because they are solved numerically. In addition, we use these soliton solutions to investigate the behavior of the soliton-soliton collisions. The importance of the artificial boundary method in applications motivates many challenging and interesting topics, such as the computation of hypersingular integrals, the boundary condition on the multi-dimensional and complex domain, and the nonlinear artificial boundary conditions that is also studied in this dissertation. For these issues, some techniques have been widely applied in practical engineering and scientific computations, whereas the others are just with some insights for further development. Due to the complexity of the nonlinear effect, it is very difficult to propose a general approach to design a suited artificial boundary condition. Therefore, further research in this topic is in urgent demand for accurate computation of the solution in nonlinear problems.