T. Ergenc, B, Karasozen, S, Piskarev

We present an analysis of the discretization methods for solving in Banach space $E$ the semilinear Cauchy problem $$ u''(t) = A u(t) + f(t,u(t)) , u(0) = u^0, u'(0) = u^1, $$ with operator $A$, which generate cosine operator function. We consider the existence of the solution in the case of compact %resolvent of operator $A$ function $f$ and give analysis of semidiscretization on a general approximation scheme, which include finite differences and projective methods. The investigation is based on the concept of the compact convergence of operators by Vainikko and in particular follows the investigations in [11] for the first order case. In order to obtain estimates for the rate of convergence it is natural to make use of conditions for the smoothness of the function $f$.
 

[11] Sandefur J. T. JR., Existence and uniqueness of solutions of second order nonlinear equations, SIAM J. Math. Anal. , 14, 477-487,  (1984)

Keywords: Second order Cauchy problem, cosine operator function, general approximation scheme, compact convergence