Solution of heat equation by a novel implicit scheme using block hybrid preconditioning of the conjugate gradient method
The main goal of the study is the approximation of the solution to the Dirichlet boundary value problem (DBVP) of the heat equation on a rectangle by developing a new difference method on a grid system of hexagons. It is proved that the given special scheme is unconditionally stable and con...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Academician Ye.A. Buketov Karaganda University
2023-03-01
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| Series: | Қарағанды университетінің хабаршысы. Математика сериясы |
| Online Access: | https://mathematics-vestnik.ksu.kz/apart/2023-109-1/07.pdf |
| _version_ | 1797843602697617408 |
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| author | S.C. Buranay N. Arshad |
| author_facet | S.C. Buranay N. Arshad |
| author_sort | S.C. Buranay |
| collection | DOAJ |
| description | The main goal of the study is the approximation of the solution to the Dirichlet boundary value problem (DBVP) of the heat equation on a rectangle by developing a new difference method on a grid system of hexagons. It is proved that the given special scheme is unconditionally stable and converges to the exact solution on the grids with fourth order accuracy in space variables and second order accuracy in time variable. Secondly, an incomplete block factorization is given for symmetric positive definite block tridiagonal (SPD-BT) matrices utilizing a conservative iterative method that approximates the inverse of the pivoting diagonal blocks by preserving the symmetric positive definite property. Subsequently, by using this factorization block hybrid preconditioning of the conjugate gradient (BHP-CG) method is applied to solve the obtained algebraic system of equations at each time level. |
| first_indexed | 2024-04-09T17:07:59Z |
| format | Article |
| id | doaj.art-d2a82e6c1e2e461ba9b121e5a1fb7861 |
| institution | Directory Open Access Journal |
| issn | 2518-7929 2663-5011 |
| language | English |
| last_indexed | 2024-04-09T17:07:59Z |
| publishDate | 2023-03-01 |
| publisher | Academician Ye.A. Buketov Karaganda University |
| record_format | Article |
| series | Қарағанды университетінің хабаршысы. Математика сериясы |
| spelling | doaj.art-d2a82e6c1e2e461ba9b121e5a1fb78612023-04-20T10:42:14ZengAcademician Ye.A. Buketov Karaganda UniversityҚарағанды университетінің хабаршысы. Математика сериясы2518-79292663-50112023-03-011091588010.31489/2023M1/58-80Solution of heat equation by a novel implicit scheme using block hybrid preconditioning of the conjugate gradient methodS.C. BuranayN. Arshad The main goal of the study is the approximation of the solution to the Dirichlet boundary value problem (DBVP) of the heat equation on a rectangle by developing a new difference method on a grid system of hexagons. It is proved that the given special scheme is unconditionally stable and converges to the exact solution on the grids with fourth order accuracy in space variables and second order accuracy in time variable. Secondly, an incomplete block factorization is given for symmetric positive definite block tridiagonal (SPD-BT) matrices utilizing a conservative iterative method that approximates the inverse of the pivoting diagonal blocks by preserving the symmetric positive definite property. Subsequently, by using this factorization block hybrid preconditioning of the conjugate gradient (BHP-CG) method is applied to solve the obtained algebraic system of equations at each time level.https://mathematics-vestnik.ksu.kz/apart/2023-109-1/07.pdf |
| spellingShingle | S.C. Buranay N. Arshad Solution of heat equation by a novel implicit scheme using block hybrid preconditioning of the conjugate gradient method Қарағанды университетінің хабаршысы. Математика сериясы |
| title | Solution of heat equation by a novel implicit scheme using block hybrid preconditioning of the conjugate gradient method |
| title_full | Solution of heat equation by a novel implicit scheme using block hybrid preconditioning of the conjugate gradient method |
| title_fullStr | Solution of heat equation by a novel implicit scheme using block hybrid preconditioning of the conjugate gradient method |
| title_full_unstemmed | Solution of heat equation by a novel implicit scheme using block hybrid preconditioning of the conjugate gradient method |
| title_short | Solution of heat equation by a novel implicit scheme using block hybrid preconditioning of the conjugate gradient method |
| title_sort | solution of heat equation by a novel implicit scheme using block hybrid preconditioning of the conjugate gradient method |
| url | https://mathematics-vestnik.ksu.kz/apart/2023-109-1/07.pdf |
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