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

TL;DR

This paper analyzes the stability and convergence of a three-layer semi-discrete scheme for an abstract nonlinear evolution equation, generalizing the Ball integro-differential equation, with proven boundedness, stability, and error estimates.

Contribution

It introduces a novel three-layer semi-discrete scheme for a nonlinear evolution equation and proves its stability, boundedness, and convergence with error estimates.

Findings

Solution and its derivative are uniformly bounded.

High-order a priori estimates are obtained for the linear problem.

The iterative method converges for each time step.

Abstract

We consider the Cauchy problem for a second-order nonlinear evolution equation in a Hilbert space. This equation represents the abstract generalization of the Ball integro-differential equation. The general nonlinear case with respect to terms of the equation which include a square of a norm of a gradient is considered. A three-layer semi-discrete scheme is proposed in order to find an approximate solution. In this scheme, the approximation of nonlinear terms that are dependent on the gradient is carried out by using an integral mean. We show that the solution of the nonlinear discrete problem and its corresponding difference analogue of a first-order derivative is uniformly bounded. For the solution of the corresponding linear discrete problem, it is obtained high-order a priori estimates by using two-variable Chebyshev polynomials. Based on these estimates we prove the stability of the nonlinear discrete problem. For smooth solutions, we provide error estimates for the approximate solution. An iteration method is applied in order to find an approximate solution for each temporal step. The convergence of the iteration process is proved.