An unconditionally stable finite difference scheme systems described by second order partial differential equations

An unconditionally stable finite difference scheme for systems whose dynamics are described by a second-order partial differential equation is developed. The scheme is motivated by the well-known Crank-Nicolson discretization which was developed for first-order systems. The stability of the finite-difference scheme is analysed by von Neumann´s method. Using the new scheme, a discrete in time and space model of a deformable mirror is derived as the basis for control law design. The convergence of this scheme for various values of the discretization parameters is checked by numerical simulations.