A class of weighted energy-preserving Du Fort-Frankel difference schemes for solving sine-Gordon-type equations

Recently, invariant energy-quadratization methods (IEQMs) have been introduced by Xiaofeng Yang's group to develop linear and energy-dissipation-preserving methods for nonlinear energy-dissipation systems. Following their work, two auxiliary functions are firstly introduced to rewrite the sine-Gordon equation (SGE) and coupled sine-Gordon equations (CSGEs) into equivalent systems, respectively. Then, two energy-preserving Du Fort-Frankel-type finite difference methods (EP-DFFT-FDMs) have been suggested for them, respectively. By using the discrete energy methods, the discrete energy conservative laws and convergence rates in the H1-norm have been derived, rigorously. It is worth mentioning that the proposed discrete energy is an approximation to the exact energy of the continuous problem. As hx = O( increment t), hy = O( increment t) and parameter lambda > 1/4, the current methods are stable in the H1-norm because numerical solutions obtained by them are bounded in the H1-norm. What is more, as parameter lambda > 1/4, the current methods are unconditionally stable in the L2-norm because numerical solutions obtained by them are uniformly bounded in the L2-norm. Moreover, our methods are explicit, and very easy to be implemented. However, a shortcoming of the current increment t increment t methods is that they are conditionally consistent. Namely, and hx tend to zero hy as time step increment t, spatial mesh sizes hx in x-direction and hy in y-direction tend to zero. Numerical findings support the correctness of theoretical analyses and the performance of the algorithms.(c) 2022 Elsevier B.V. All rights reserved.