A conservative fully discrete finite element scheme for the nonlinear Klein–Gordon equation
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...
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| Format: | Article |
| Language: | English |
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Kazan Federal University
2024-01-01
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| Series: | Учёные записки Казанского университета. Серия Физико-математические науки |
| Subjects: | |
| Online Access: | https://uzakufismat.elpub.ru/jour/article/view/10 |
| _version_ | 1797205454991917056 |
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| author | R. Z. Dautov G. R. Salimzyanova |
| author_facet | R. Z. Dautov G. R. Salimzyanova |
| author_sort | R. Z. Dautov |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-07T23:47:32Z |
| format | Article |
| id | doaj.art-c31798cf00cc44d5831964d8eb251cab |
| institution | Directory Open Access Journal |
| issn | 2541-7746 2500-2198 |
| language | English |
| last_indexed | 2024-04-24T08:51:23Z |
| publishDate | 2024-01-01 |
| publisher | Kazan Federal University |
| record_format | Article |
| series | Учёные записки Казанского университета. Серия Физико-математические науки |
| spelling | doaj.art-c31798cf00cc44d5831964d8eb251cab2024-04-16T11:42:17ZengKazan Federal UniversityУчёные записки Казанского университета. Серия Физико-математические науки2541-77462500-21982024-01-01165319020710.26907/2541-7746.2023.3.190-2079A conservative fully discrete finite element scheme for the nonlinear Klein–Gordon equationR. Z. Dautov0G. R. Salimzyanova1Kazan Federal UniversityKazan Federal UniversityThis 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.https://uzakufismat.elpub.ru/jour/article/view/10petrov–galerkin methodfinite element methodklein–gordon equationimplicit scheme |
| spellingShingle | R. Z. Dautov G. R. Salimzyanova A conservative fully discrete finite element scheme for the nonlinear Klein–Gordon equation Учёные записки Казанского университета. Серия Физико-математические науки petrov–galerkin method finite element method klein–gordon equation implicit scheme |
| title | A conservative fully discrete finite element scheme for the nonlinear Klein–Gordon equation |
| title_full | A conservative fully discrete finite element scheme for the nonlinear Klein–Gordon equation |
| title_fullStr | A conservative fully discrete finite element scheme for the nonlinear Klein–Gordon equation |
| title_full_unstemmed | A conservative fully discrete finite element scheme for the nonlinear Klein–Gordon equation |
| title_short | A conservative fully discrete finite element scheme for the nonlinear Klein–Gordon equation |
| title_sort | conservative fully discrete finite element scheme for the nonlinear klein gordon equation |
| topic | petrov–galerkin method finite element method klein–gordon equation implicit scheme |
| url | https://uzakufismat.elpub.ru/jour/article/view/10 |
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