Energy-preserving Du Fort-Frankel difference schemes for solving sine-Gorden equation and coupled sine-Gordon equations
Energy-preserving Du Fort-Frankel difference schemes for solving sine-Gorden equation and coupled sine-Gordon equations
复制标题
求解正弦-Gordon方程和耦合正弦-Gordon方程的能量守恒Du Fort-Frankel差分格式
DOI：
10.1007/s11075-022-01453-1
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发表时间：
--
影响因子：
2.1
通讯作者：
Qihong Wang
中科院分区：
文献类型：
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作者：
Dingwen Deng;Jingliang Chen;Qihong Wang
DuFort-Frankel（DFF）有限差分方法（FDM）是由DuFort和Frankel于1953年提出的求解具有周期边界条件的线性扩散方程的方法。它是一个显式的无条件冯诺依曼稳定格式。因此，它是非常容易实现的，适合于长期的模拟。然而，目前还没有关于用能量守恒的Du Fort-Frankel有限差分方法（EP-DFF-FDM）数值求解sine-Gordon方程（SGE）和非线性耦合sine-Gordon方程（CSGE）的研究。在这项研究中，两类加权的EP-DFF-FDM，这是设计相结合的DFF FDM与不变能量二次化方法（IEQM），分别为SGE和CSGE的数值模拟。利用离散能量方法，证明了它们的解满足离散能量守恒律，并收敛到H ~ 1-范数阶的精确解。这里，τ表示时间增量，而x和h分别表示x和y维的间隔网格。此外，当参数λ≥ 1/4时，我们的方法虽然是显格式，但在L2范数下是无条件稳定的。最后，数值结果证实了理论研究结果的正确性，以及我们的算法优于一些现有的算法在计算效率和能力，以保持离散能量。
Du Fort-Frankel (DFF) finite difference method (FDM) was proposed for linear diffusion equations with periodic boundary conditions by Du Fort and Frankel in 1953. It is an explicit and unconditionally von Neumann stable scheme. Thus, it is very easy to be implemented and suitable for long-term simulations. However, there has been no research work on numerical solutions of sine-Gordon equations (SGE) and nonlinear coupled sine-Gordon equations (CSGEs) by using energy-preserving Du Fort-Frankel finite difference methods (EP-DFF-FDMs). In this study, two classes of weighted EP-DFF-FDMs, which are devised by combining DFF FDMs with invariant energy quadratization methods (IEQMs), are suggested for numerical simulations of SGE and CSGEs, respectively. By using the discrete energy method, it is shown that their solutions satisfy the discrete energy conservative laws, and converge to exact solutions with an order ofinH1-norm. Here,τdenotes time increment, whilehxandhyrepresent spacing grids inx- andy-dimensions, respectively. What is more, our methods with parameter𝜃≥ 1/4 are unconditionally stable inL2-norm though they are explicit schemes. Finally, numerical results confirm the exactness of theoretical findings, and the superiorities of our algorithms over some existent algorithms in terms of computational efficiency and the ability to conserve the discrete energy.