On local convergence of a symmetric semi-discrete scheme for an abstract analogue of the Kirchhoff equation

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.

On convergence of a three‐layer semi‐discrete scheme for the non‐linear dynamic string equation of Kirchhoff‐type with time‐dependent coefficients

“…The works authored by Rogava and Tsiklauri [33,34] are primarily concerned with the development and analysis of a symmetric three-layer semi-discrete scheme for solving the Cauchy problem associated with the abstract generalization of the dynamic Kirchhoff equation. In the proposed scheme, the value of a non-linear term is evaluated at the middle node point, resulting in the transformation of the original problem into a linear one for each temporal layer.…”

“…Refs. [33][34][35][36]). By applying this approach, the stated hyperbolic non-linear partial differential equation can be reduced to a system of linear ordinary differential equations of second order.…”

“…To maintain the self-contained nature of this paper, we state the following lemma along with a reference to its proof (see Ref. [33,Lemma 3.2]). Lemma 3.3 see the Lemma 3.2 in Ref.…”

“…Lemma 3.3 see the Lemma 3.2 in Ref. [33]. Let the sequences of non-negative numbers, {𝛼 𝑘 } 𝑛 𝑘=0 and {𝑐 𝑘 } 𝑛 𝑘=0 , be such that they fulfil the following inequality:…”

“…It should be noted that the order of accuracy for the semi-discrete scheme (2.2), when combined with the spatial discretization algorithm, is (𝜏 2 + ℎ 4 ).Consider the following pair of test problems (cf. Refs [33,35,36]…”