Convergence of a semi-discrete scheme for an abstract nonlinear second order evolution equation

In the paper there is considered the Cauchy problem for an abstract nonlinear second order evolution equation in the Hilbert space. This equation represents a generalization of a nonlinear Kirchhoff-type beam equation. For approximate solution of this problem, we introduce a three-layer semi-discrete scheme, where the value of the gradient in the nonlinear term is taken at the middle point. This makes possible to reduce the finding of the approximate solution on each time step to solution of the linear problem. It is proved that the solution of the nonlinear discrete problem, as well as its corresponding difference analog of the first order derivative, is uniformly bounded. For the corresponding linear discrete problem, the high order a priori estimates are obtained using classic Chebyshev polynomials. Based on these facts, for nonlinear discrete problem, the a priori estimates are proved, whence the stability and error estimates of the approximate solution follow. Using the constructed scheme, numerical calculations for various test problems are carried out.