Error estimates of the cubic interpolated pseudo-particle scheme for one-dimensional advection equations
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916411298807808 |
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| author | Kashiwabara, Takahito Takemura, Haruki |
| author_facet | Kashiwabara, Takahito Takemura, Haruki |
| contents | Error estimates of cubic interpolated pseudo-particle scheme (CIP scheme) for the one-dimensional advection equation with periodic boundary conditions are presented. The CIP scheme is a semi-Lagrangian method involving the piecewise cubic Hermite interpolation. Although it is numerically known that the space-time accuracy of the scheme is third order, its rigorous proof remains an open problem. In this paper, denoting the spatial and temporal mesh sizes by $ h $ and $ Δt $ respectively, we prove an error estimate $ O(Δt^3 + \frac{h^4}{Δt}) $ in $ L^2 $ norm theoretically, which justifies the above-mentioned prediction if $ h = O(Δt) $. The proof is based on properties of the interpolation operator; the most important one is that it behaves as the $ L^2 $ projection for the second-order derivatives. We remark that the same strategy perfectly works as well to address an error estimate for the semi-Lagrangian method with the cubic spline interpolation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11885 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Error estimates of the cubic interpolated pseudo-particle scheme for one-dimensional advection equations Kashiwabara, Takahito Takemura, Haruki Numerical Analysis 65M06 Error estimates of cubic interpolated pseudo-particle scheme (CIP scheme) for the one-dimensional advection equation with periodic boundary conditions are presented. The CIP scheme is a semi-Lagrangian method involving the piecewise cubic Hermite interpolation. Although it is numerically known that the space-time accuracy of the scheme is third order, its rigorous proof remains an open problem. In this paper, denoting the spatial and temporal mesh sizes by $ h $ and $ Δt $ respectively, we prove an error estimate $ O(Δt^3 + \frac{h^4}{Δt}) $ in $ L^2 $ norm theoretically, which justifies the above-mentioned prediction if $ h = O(Δt) $. The proof is based on properties of the interpolation operator; the most important one is that it behaves as the $ L^2 $ projection for the second-order derivatives. We remark that the same strategy perfectly works as well to address an error estimate for the semi-Lagrangian method with the cubic spline interpolation. |
| title | Error estimates of the cubic interpolated pseudo-particle scheme for one-dimensional advection equations |
| topic | Numerical Analysis 65M06 |
| url | https://arxiv.org/abs/2402.11885 |