具有Neumann边界条件的sine-Gordon方程的有效保能量数值逼近
Efficient energy-preserving numerical approximations for the sine-Gordon equation with Neumann boundary conditions
摘要：
我们提出了两个新的类的全离散能量保持算法的sine-Gordon方程受到诺依曼边界条件。首先利用余弦准谱方法在两种不同的网格下进行保结构的空间离散，得到两个有限维Hamilton常微分方程组。然后，我们结合预测校正Crank-Nicolson计划的投影方法，达到完全离散的能量保持方法。或者，我们引入一个补充变量，将初始模型转换成一个松弛系统，这使我们能够更容易地构造结构保持算法。在空间上采用余弦准谱方法，在时间上采用预测校正Crank-Nicolson格式，直接离散松弛方程组，得到一类新的能量守恒格式.提出的方法可以有效地解决离散余弦变换。一些基准的例子和数值比较证明所提出的计划的精度，效率和优越性。
英文摘要：
We present two novel classes of fully discrete energy-preserving algorithms for the sine-Gordon equation subject to Neumann boundary conditions. The cosine pseudo-spectral method is first used to develop structure-preserving spatial discretizations under two different meshes, which result two finite-dimensional Hamiltonian ODE systems. Then we combine the prediction-correction Crank-Nicolson scheme with the projection approach to arrive at fully discrete energy-preserving methods. Alternatively, we introduce a supplementary variable to transform the initial model into a relaxation system, which allows us to construct structure-preserving algorithms more easily. We then discretize the relaxation system directly by using the cosine pseudo-spectral method in space and the prediction-correction Crank-Nicolson scheme in time to derive a new class of energy-preserving schemes. The proposed methods can be solved effectively by the discrete Cosine transform. Some benchmark examples and numerical comparisons are presented to demonstrate the accuracy, efficiency and superiority of the proposed schemes.
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