基于IEQ方法的分数阶sine-Gordon方程的线性隐式保结构格式
A linearly implicit structure-preserving scheme for the fractional sine-Gordon equation based on the IEQ approach
摘要：
基于新发展的不变能量二次化方法，提出了求解空间分数阶sine-Gordon方程的一种保结构线性隐式数值格式.首先，我们利用分数拉普拉斯泛函的变分导数将方程转化为正则哈密顿系统。然后，利用分数阶中心差分公式对由不变能量二次化方法得到的等效系统进行空间方向的离散，得到一个守恒的半离散格式。随后，线性隐式结构保持方法应用于所得到的半离散系统，以达到一个全离散的保守格式。给出了数值格式的稳定性、可解性和在最大范数下的收敛性。在实际计算中，采用基于快速傅立叶变换的快速算法，降低了计算复杂性。最后，数值例子来证实我们的理论分析结果。
英文摘要：
This paper aims to develop a linearly implicit structure-preserving numerical scheme for the space fractional sine-Gordon equation, which is based on the newly developed invariant energy quadratization method. First, we reformulate the equation as a canonical Hamiltonian system by virtue of the variational derivative of the functional with fractional Laplacian. Then, we utilize the fractional centered difference formula to discrete the equivalent system derived by the invariant energy quadratization method in space direction, and obtain a conservative semi-discrete scheme. Subsequently, the linearly implicit structure-preserving method is applied for the resulting semi-discrete system to arrive at a fully-discrete conservative scheme. The stability, solvability and convergence in the maximum norm of the numerical scheme are given. Furthermore, a fast algorithm based on the fast Fourier transformation technique is used to reduce the computational complexity in practical computation. Finally, numerical examples are provided to confirm our theoretical analysis results.
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