论文标题
具有诺伊曼边界条件的正弦格式方程的有效节能的数值近似值
Efficient energy-preserving numerical approximations for the sine-Gordon equation with Neumann boundary conditions
论文作者
论文摘要
我们提供了两种新型的针对正弦 - 戈登方程的完全离散能量的算法，但受诺伊曼边界条件的约束。余弦伪谱法首先用于在两个不同的网格下开发具有结构性的空间离散化，从而导致两个有限的二维Hamiltonian Ode Systems。然后，我们将预测 - 校正曲柄 - 尼科尔森方案与投影方法相结合，以完全离散的能量保存方法。另外，我们引入了一个补充变量，以将初始模型转换为放松系统，这使我们能够更容易构造结构传播算法。然后，我们通过在空间中使用余弦伪 - 光谱法直接离散放松系统，并及时使用预测校正曲柄 - 尼科尔森方案来得出一类新的能量保存方案。所提出的方法可以通过离散的余弦变换有效地解决。提供了一些基准示例和数值比较，以证明所提出的方案的准确性，效率和优越性。
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.
