Parameters spline methods for the solution of hyperbolic equations

In this paper, by using a parameters cubic spline in space and compact finite difference in time direction, we get a class of finite difference schemes for solving second-order hyperbolic equations with mixed boundary conditions. Stability analysis of the methods have been carried out. It has been shown that by suitable choosing the cubic spline parameters most of the previous known methods for homogeneous and non-homogeneous cases can be derived from our methods. We also obtain new high accuracy schemes of O(h2+τ2h2)O(h2+τ2h2) and O(h4+τ2h4)O(h4+τ2h4). Numerical comparison with Rashidinia’s method [J. Rashidinia et al., Spline methods for the solutions of hyperbolic equations, Appl. Math. Comput. 190 (2007) 882–886] shows the superiority of our presented schemes.