Convergence Analysis of a Fully Discrete Observer For Data Assimilation of the Barotropic Euler Equations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912962883616768 |
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| author | Chaumet, Aidan Giesselmann, Jan |
| author_facet | Chaumet, Aidan Giesselmann, Jan |
| contents | We study the convergence of a discrete Luenberger observer for the barotropic Euler equations in one dimension, for measurements of the velocity only. We use a mixed finite element method in space and implicit Euler integration in time. We use a modified relative energy technique to show an error bound comparing the discrete observer to the original system's solution. The bound is the sum of three parts: an exponentially decaying part, proportional to the difference in initial value, a part proportional to the grid sizes in space and time and a part that is proportional to the size of the measurement errors as well as the nudging parameter. The proportionality constants of the second and third parts are independent of time and grid sizes. To the best of our knowledge, this provides the first error estimate for a discrete observer for a quasilinear hyperbolic system, and implies uniform-in-time accuracy of the discrete observer for long-time simulations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_10962 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Convergence Analysis of a Fully Discrete Observer For Data Assimilation of the Barotropic Euler Equations Chaumet, Aidan Giesselmann, Jan Numerical Analysis Optimization and Control 35L65, 93B53, 93C20, 93C41, 65M15, 76M10 We study the convergence of a discrete Luenberger observer for the barotropic Euler equations in one dimension, for measurements of the velocity only. We use a mixed finite element method in space and implicit Euler integration in time. We use a modified relative energy technique to show an error bound comparing the discrete observer to the original system's solution. The bound is the sum of three parts: an exponentially decaying part, proportional to the difference in initial value, a part proportional to the grid sizes in space and time and a part that is proportional to the size of the measurement errors as well as the nudging parameter. The proportionality constants of the second and third parts are independent of time and grid sizes. To the best of our knowledge, this provides the first error estimate for a discrete observer for a quasilinear hyperbolic system, and implies uniform-in-time accuracy of the discrete observer for long-time simulations. |
| title | Convergence Analysis of a Fully Discrete Observer For Data Assimilation of the Barotropic Euler Equations |
| topic | Numerical Analysis Optimization and Control 35L65, 93B53, 93C20, 93C41, 65M15, 76M10 |
| url | https://arxiv.org/abs/2603.10962 |