【摘要】 本文对Neumann边界条件下的sine-Gordon方程提出两类新的全离散高效保能量算法.首先考虑在两种不同空间网格上应用cosine拟谱方法去发展空间保结构格式,导出两个有限维Hamilton常微分方程系统.然后,将预估校正型的Crank-Nicolson格式和投影方法相结合,得到一类全离散保能量算法.另外,本文对sine-Gordon方程引入一个补充变量,将原始模型转化成一个松弛系统,这使得保结构算法更容易被发展.本文针对等价的松弛系统仍采用cosine拟谱方法和预估校正的CrankNicolson格式进行离散,发展了另一类新的保能量算法.本文提出的数值格式不仅保持系统的原始能量,而且可以通过离散cosine变换进行高效快速求解.最后,数值实验验证了格式的数值精度、计算效率和优秀性态.更多还原

【Abstract】 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 firstly used to develop structure-preserving spatial discretizations under two different meshes, which result in two finite-dimensional Hamiltonian ODE(ordinary differential equation) systems. Then we combine the prediction-correction CrankNicolson 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 develop 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 efficiently by the discrete cosine transform. Some benchmark examples and numerical comparisons are presented to demonstrate the accuracy,efficiency and superiority of the proposed schemes.更多还原