Numerical solution of one-dimensional Sine–Gordon equation using high accuracy multiquadric quasi-interpolation

In this paper, we propose a numerical scheme to solve one-dimensional Sine–Gordon equation related to many scientific research topics by using high accuracy multiquadric quasi-interpolation. We use the derivatives of a multiquadric quasi-interpolant to approximate the spatial derivatives, and a finite difference to approximate the temporal derivative. The advantages of the scheme are that it is meshfree, and in each time step only a multiquadric quasi-interpolant is employed, so that the algorithm is very easy to implement. The accuracy of our scheme is demonstrated by some test problems.