摘要
对微分方程数值解法的研究不仅是计算数学的重要内容，而且在其它学科领域也具有广泛应用，如计算物理、化学、生物等。在现实世界中，绝大部分问题所对应的数学模型是非线性微分方程，其中之一就是非线性Schr(o|¨)dinger（NLS）方程。NLS方程在物理学的很多领域都发挥着重要作用。所以，本文围绕NLS方程及其相关问题用数值方法进行求解和研究。本文主要采用的数值方法有正交样条配置（OSC）方法、有限差分法和分裂步方法等。
本文首先为带波动算子的NLS方程和耦合非线性Klein-Gordon-Schr(o|¨)dinger方程构造了时间离散OSC方法。从理论上严格证明了格式的守恒性、收敛性和稳定性，并用数值实验检验了理论结果。
分裂步方法能与有限差分法、谱方法、有限元方法等相结合形成有效的数值方法，但是分裂步方法与OSC方法相结合的工作却没有找到。所以，本文将这两种方法结合起来形成新的数值方法，称为分裂步OSC（SSOSC）方法。将此方法用于数值求解一维耦合NLS方程，并与其它三种时间离散OSC格式相比较，发现SSOSC方法最有效。如此，分裂步方法与OSC方法的结合是可行的。
鉴于SSOSC方法的可信性和有效性，此方法又被推广到求解多维NLS方程上。经大量数值实验的检测，本次推广也是成功的。由于用OSC方法求解三维问题的研究工作还没有发现，本文的这部分工作也算是振奋人心的。另外，考虑到NLS方程中依赖于时间的势函数有时候是不能关于时间变量精确积分的，这样就减弱了分裂步方法的优势。本文用数值积分方法解决了这个问题，而且没有降低原方法的精度阶数。值得一提的是，本文采用的数值积分方法不仅对SSOSC方法有效，对分裂步有限差分方法和分裂步Fourier谱方法也同样有效。
Ginzburg-Landau（GL）方程是比NLS方程更复杂的模型，其方程系数都是复数类型的。Kuramoto-Tsuzuki方程从形式来看是一个一维三次GL方程，本文用有限差分法来数值求解该方程，从理论上论证了非线性差分格式与线性化差分格式的关系，并通过数值算例进行验证。对二维三次GL方程，本文构造了一个分裂步有限差分格式和一个分裂步交替方向隐式格式，分析了两个格式之间的关系，用方程的平面波解来研究格式的性质，并用线性化分析方法讨论格式的稳定性。数值实验检测了格式的数值性能。对多维三次-五次GL（CQGL）方程，本文用分裂步紧有限差分（SSCFD）方法来数值求解。首先，将CQGL方程分裂成两个非线性子问题和一个或多个线性子问题，用紧有限差分格式求解线性子问题。由于非线性子问题不能像通常情况那样直接精确积分求解，导致常规的SSCFD方法失效。在此情形下，本文采用Runge-Kutta方法来数值求解，但并未降低原方法的精度阶数。经大量数值实验验证这种数值方法是可行且有效的。即使是对那些非线性子问题可以精确求解的情形，本文的数值方法仍能表现得与常规SSCFD方法一样好。这也说明了非线性子问题必须精确求解这个要求并不总是必须的。这是对分裂步方法的推广和发展。
The study on numerical methods for partial differential equations is not only an important branch of computational mathematics, but also has extensive applications in many other fields, such as computational physics, chemistry, biology, et al. Most models describing various phenomena in the world are nonlinear differential equations. One of them is the nonlinear Schrodinger (NLS) equation which plays a key role in many fields of physics. Therefore, some nonlinear Schrodinger equations and their related issues are numerically resolved and studied in this thesis. The main numerical methods applied are the orthogonal spline collocation (OSC) method, the finite difference method, and the split step technique.
Firstly, discrete-time OSC schemes are proposed for the NLS equation with wave operator and the coupled nonlinear Klein-Gordon-Schrodinger equation. The conservation, convergence and stability of the schemes are strictly proved in theory. Numerical experiments are carried out to verify the theorical results.
The split step method is efficiently combined with the finite difference method, the spectral method, or the finite element method and so on. Surprisingly, the combination of the split step method and the OSC method is missed. Therefore, the combination is tried in this thesis for the coupled NLS equation in one dimension. The new method is named as split-step OSC (SSOSC) method by the author. The SSOSC method is compared with three other discrete-time OSC schemes, and the new method is superior. Thus, the first step is successful.
As the SSOSC method is reliable and efficient, it is extended to multidimensional NLS equations. And this extension is also successful, which is examed by many numerical tests. Since no research on three-dimensional problems by the OSC method has been found, this step is really inspiring. Moreover, sometimes the time-dependent potential in the NLS equation cannot be integrated exactly which weakens the advantage of the split step method, so numerical integration is utilized by the author. The approximate strategy is efficient and does not reduce the original accuracy. It is worth noting that this approximation is efficient not only for the SSOSC method, but also for the split-step finite difference method and the split-step Fourier spectral method.
Ginzburg-Landau (GL) equations are more complicated than the NLS ones. Finite difference schemes are presented for the Kuramoto-Tsuzuki equation which is a one-dimensional cubic GL equation in form. The relation between the nonlinear schemes and the linearized ones is discussed in theory, and also verified by numerical examples. For the two-dimensional cubic GL equation, a split-step finite difference (SSFD) scheme and a split-step alternating direction implicit (ADI) scheme are proposed. The connection between the two schemes is discussed, and the plane wave solution of the equation is used to analyze the schemes. Linearized analysis is applied to discuss the stability, and numerical tests are also carried out to examine the schemes. Split-step compact finite difference schemes are constructed for the multi-dimensional cubic-quintic GL (CQ GL) equations. Firstly, the split step method is applied, and the CQ GL equations are separated into two nonlinear subproblems and one or several linear ones. The linear subproblems are resolved by the compact finite difference schemes. As the nonlinear subproblems cannot be integrated exactly as usual, the normal split-step compact finite difference (SSCFD) method is failed. Theorefore, the author utilizes the Runge-Kutta method, and the accuracy order is not reduced. Extensive numerical experiments are carried out to show that the proposed numerical approximation is successful and efficient. And even for the situations that the nonlinear subproblems can be integrated exactly, the present method is still as good as the usual SSCFD one. It implies that the requirement of integrating the nonlinear subproblems exactly may be not always necessary. This is an extension and improvement of the split step method.
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