Higher order scheme for two-dimensional inhomogeneous sine-Gordon equation with impulsive forcing

In this paper higher order scheme is presented for two-dimensional sine-Gordon equation. Higher order Legendre spectral element method is used for space discretization which is basically a domain decomposition method and it retains all the advantages of spectral and finite element methods. Spectral stability analysis is performed for both homogeneous and inhomogeneous sine-Gordon equations which gives implicit expressions of critical time step. Various test cases are solved which shows the robustness and accuracy of the proposed scheme. Moreover, experimental order of convergence is obtained which gives optimal convergence rate. It is also shown that the proposed scheme exhibit conservation of energy for undamped sine-Gordon equation. In realistic scenario defects are predominant in nature. sine-Gordon equation with defect can be effectively modeled by additional impulsive forcing term. In the second part of this paper, proposed higher order scheme is used to solve inhomogeneous sine-Gordon equation with impulsive loading. Both constant as well as time-dependent strength of impulsive forcing are used. Soliton solution to sine-Gordon equation is analyzed under the action of such forcing. Different Dirac delta representations are discussed which can accurately replicate behavior of impulsive loading. This is important because, dynamics of soliton behavior strongly depends on the impulsive force representation. Various conclusions are made based on this study.