$hp$-version $C^1$-continuous Petrov-Galerkin method for nonlinear second-order initial value problems with application to wave equations
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866910414187528192 |
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| author | Wang, Lina Zhang, Mingzhu Tian, Hongjiong Yi, Lijun |
| author_facet | Wang, Lina Zhang, Mingzhu Tian, Hongjiong Yi, Lijun |
| contents | We introduce and analyze an $hp$-version $C^1$-continuous Petrov-Galerkin (CPG) method for nonlinear initial value problems of second-order ordinary differential equations. We derive a-priori error estimates in the $L^2$-, $L^\infty$-, $H^1$- and $H^2$-norms that are completely explicit in the local time steps and local approximation degrees. Moreover, we show that the $hp$-version $C^1$-CPG method superconverges at the nodal points of the time partition with regard to the time steps and approximation degrees. As an application, we apply the $hp$-version $C^1$-CPG method to time discretization of nonlinear wave equations. Several numerical examples are presented to verify the theoretical results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_14345 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $hp$-version $C^1$-continuous Petrov-Galerkin method for nonlinear second-order initial value problems with application to wave equations Wang, Lina Zhang, Mingzhu Tian, Hongjiong Yi, Lijun Numerical Analysis 65L60, 65L05, 65L70 We introduce and analyze an $hp$-version $C^1$-continuous Petrov-Galerkin (CPG) method for nonlinear initial value problems of second-order ordinary differential equations. We derive a-priori error estimates in the $L^2$-, $L^\infty$-, $H^1$- and $H^2$-norms that are completely explicit in the local time steps and local approximation degrees. Moreover, we show that the $hp$-version $C^1$-CPG method superconverges at the nodal points of the time partition with regard to the time steps and approximation degrees. As an application, we apply the $hp$-version $C^1$-CPG method to time discretization of nonlinear wave equations. Several numerical examples are presented to verify the theoretical results. |
| title | $hp$-version $C^1$-continuous Petrov-Galerkin method for nonlinear second-order initial value problems with application to wave equations |
| topic | Numerical Analysis 65L60, 65L05, 65L70 |
| url | https://arxiv.org/abs/2303.14345 |