An explicit difference method for the wave equation with extended stability range

Abstract

A method is developed which superimposes a uniform grid of step-sizeh on the space variablex in the wave equation∂ ²u/∂x²=∂²u/∂t². The resulting system of second order ordinary differential equations is solved using a rational approximant toe ^lA, wherel is the time step andA is the coefficient matrix. A seven point explicit finite difference scheme is derived whose consistency, stability and convergence are discussed. The rational approximant is seen to have a stability range of 0 ≦l/h=r≦√3. Numerical results of the algorithm applied to two problems, one of which has a discontinuity between the initial and boundary conditions, are reported and compared with the familiar five point explicit scheme, which may be derived using the same approach with a different rational approximant and whose stability range is 0≦r≦1.