A conservative fully discrete finite element scheme for the nonlinear Klein–Gordon equation

Autor: R. Z. Dautov, G. R. Salimzyanova
Jazyk: English<br />Russian
Rok vydání: 2024
Předmět:
Zdroj: Учёные записки Казанского университета. Серия Физико-математические науки, Vol 165, Iss 3, Pp 190-207 (2024)
Druh dokumentu: article
ISSN: 2541-7746
2500-2198
DOI: 10.26907/2541-7746.2023.3.190-207
Popis: This article proposes a family of the Petrov–Galerkin–FEM methods that can be used to solve the nonlinear Klein–Gordon equation. The discrete schemes were formulated based on the solution of the problem and its time derivative. They ensure that the total energy is conserved at a discrete level. The simplest two-layer scheme was studied numerically. Based on the solution of the test problems with smooth solutions, it was shown that the scheme can determine the solution of the problem, as well as its time derivative with an error of the order of O(h2 + τ 2) in the continuous L2 norm, where τ and h characterize the grid steps in time and space, respectively.
Databáze: Directory of Open Access Journals