Numerical solutions of a damped sine-Gordon equation in two space variables (Q1905550)

Let \(\Omega \subset \mathbb{R}^2\) be a rectangle and \(\beta \geq 0\). The authors consider the numerical solution of the damped sine-Gordon equation \[ {\partial^2 u\over \partial x^2} + {\partial^2 u\over \partial y^2} - {\partial^2 u \over \partial t^2} - \beta{\partial u\over \partial t} = F(x,y) \sin u \] in \(\Omega\) for \(t > 0\) with initial conditions and Neumann boundary conditions. Standard discretization in the space variables gives a system of ordinary differential equations. A two-step, one-parameter method is proposed for the numerical solution of this system. The local truncation error is given. The stability of the method and the dispersion error are studied. Global extrapolation in space and time is applied to improve the accuracy. Several numerical examples are considered for various cases including line and ring solitons.