Cubic Splines Difference Method for Solving the Second Order Wave Equation

In solving initial boundary value problem of the second order wave equation,the classical finite difference method is generally restricted by its stability and precision.In this paper,four classes of three-level implicit schemes are proposed for solving the one-dimensional wave equation by using cubic-spline function.Those methods are of order O(τ2+h2),O(τ2+h4),O(τ4+h2) and O(τ4+h4)respectively.Stability conditions are obtained by Fourier analysis method.It is shown by numerical examples that the two schemes presented in this paper are much better than the precise time-integration method,the classical C-N method and the high accuracy compact schemes,and they have high accuracy even for a long time calculation.In addition,the current schemes include the high accuracy compact scheme as a special case.