On Stability and Convergence of Three-layer Semi-discrete Scheme for an Abstract Analogue of the Ball Integro-differential Equation

Cauchy problem for second-order nonlinear evolution equation is considered. This equation represents the abstract generalization of the Ball integro-differential equation. The general nonlinear case concerning terms of the equation which include a square of a norm of a gradient is considered. Three-layer semi-discrete scheme is proposed for numerical computations. In this scheme, approximation of nonlinear terms that are dependent on the gradient is done using integral averaging. Here is proved that solution of the nonlinear discrete problem and its corresponding first-order difference is uniformly bounded. For the solution of corresponding linearized discrete problem high-order, a priori estimation is obtained using two-variable Chebyshev polynomials. Based on this estimation stability of the nonlinear discrete problem is shown. For smooth solutions, it is obtained estimation for the error of the approximate solution. An approximate solution for each time step we apply the iteration method. The convergence of the iteration method is proved.