Energy-preserving Du Fort-Frankel difference schemes for solving sine-Gordon equation and coupled sine-Gordon equations

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 of O(tau(2) + h(x)(2) + h(y)(2) +( tau|h(x) )(2) + ( tau|h(y) )(2)) in H-1-norm.Here, tau denotes time increment, while hx and hy represent spacing grids in x- and y-dimensions, respectively. What is more, our methods with parameter theta >= 1/4 are unconditionally stable in L-2-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.

Automated summary

In this study, two classes of weighted energy-preserving Du Fort-Frankel finite difference methods are proposed for numerical simulations of sine-Gordon equations and nonlinear coupled sine-Gordon equations. It is shown that these methods satisfy the discrete energy conservative laws and converge to exact solutions in the H-1 norm. Furthermore, our methods are unconditionally stable in the L-2 norm despite being explicit schemes. Numerical results confirm the accuracy of theoretical findings and demonstrate the superiority of our algorithms in terms of computational efficiency and energy conservation.