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

摘要

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 H-1-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 h(x) = O(Delta t), h(y) = O(.t) and parameter lambda >= 1/4, the current methods are stable in the H-1-norm because numerical solutions obtained by them are bounded in the H-1-norm. What is more, as parameter lambda >= 1/4, the current methods are unconditionally stable in the L-2-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 methods is that they are conditionally consistent. Namely del(t)/h(x) and Delta(t)/h(y) tend to zero as time step Delta t, spatial mesh sizes h(x) in x-direction and h(y) in y-direction tend to zero. Numerical findings support the correctness of theoretical analyses and the performance of the algorithms.

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