Unconditionally optimal error estimate of a linearized variable-time-step BDF2 scheme for nonlinear parabolic equations
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2022
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866911125816213504 |
|---|---|
| author | Zhao, Chengchao Liu, Nan Ma, Yuheng Zhang, Jiwei |
| author_facet | Zhao, Chengchao Liu, Nan Ma, Yuheng Zhang, Jiwei |
| contents | In this paper we consider a linearized variable-time-step two-step backward differentiation formula (BDF2) scheme for solving nonlinear parabolic equations. The scheme is constructed by using the variable time-step BDF2 for the linear term and a Newton linearized method for the nonlinear term in time combining with a Galerkin finite element method (FEM) in space. We prove the unconditionally optimal error estimate of the proposed scheme under mild restrictions on the ratio of adjacent time-steps, i.e. $0<r_k < r_{\max} \approx 4.8645$ and on the maximum time step. The proof involves the discrete orthogonal convolution (DOC) and discrete complementary convolution (DCC) kernels, and the error splitting approach. In addition, our analysis also shows that the first level solution $u^1$ obtained by BDF1 (i.e. backward Euler scheme) does not cause the loss of global accuracy of second order. Numerical examples are provided to demonstrate our theoretical results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_06008 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Unconditionally optimal error estimate of a linearized variable-time-step BDF2 scheme for nonlinear parabolic equations Zhao, Chengchao Liu, Nan Ma, Yuheng Zhang, Jiwei Numerical Analysis 65M06, 65M12 In this paper we consider a linearized variable-time-step two-step backward differentiation formula (BDF2) scheme for solving nonlinear parabolic equations. The scheme is constructed by using the variable time-step BDF2 for the linear term and a Newton linearized method for the nonlinear term in time combining with a Galerkin finite element method (FEM) in space. We prove the unconditionally optimal error estimate of the proposed scheme under mild restrictions on the ratio of adjacent time-steps, i.e. $0<r_k < r_{\max} \approx 4.8645$ and on the maximum time step. The proof involves the discrete orthogonal convolution (DOC) and discrete complementary convolution (DCC) kernels, and the error splitting approach. In addition, our analysis also shows that the first level solution $u^1$ obtained by BDF1 (i.e. backward Euler scheme) does not cause the loss of global accuracy of second order. Numerical examples are provided to demonstrate our theoretical results. |
| title | Unconditionally optimal error estimate of a linearized variable-time-step BDF2 scheme for nonlinear parabolic equations |
| topic | Numerical Analysis 65M06, 65M12 |
| url | https://arxiv.org/abs/2201.06008 |