Neumann边界条件下sine-Gordon方程的高效保能量算法

Efficient energy-preserving numerical approximations for the sine-Gordon equation with Neumann boundary conditions

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.

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.