Numerical simulation of the N-dimensional sine-Gordon equation via operational matrices
In this paper, we develop a numerical method for the N-dimensional sine-Gordon equation using differentiation matrices, in the theoretical frame of matrix differential equations. Our method avoids calculating exponential matrices, is very intuitive and easy to express, and can be implemented without toil in any number of spatial dimensions. Although there is currently a vast literature on the numerical treatment of the one-dimensional sine-Gordon equation, the references for the two-dimensional case are much sparser, and virtually nonexistent for higher dimensions. We apply it to a battery of two-dimensional problems taken from the literature, showing that it largely outperforms the previously existing algorithms; while for three-dimensional problems, the results seem very promising.
sine-gordon equation matrix differential equations pseudo-spectral methods integrating factor conservation of energy