A Non-Conservative, Non-Local Approximation of the Burgers Equation
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917196951715840 |
|---|---|
| author | Ghoshal, Shyam Sundar Venkatesh, Parasuram Wiedemann, Emil |
| author_facet | Ghoshal, Shyam Sundar Venkatesh, Parasuram Wiedemann, Emil |
| contents | The analysis of non-local regularisations of scalar conservation laws is an active research program. Applications of such equations are found in the modelling of physical phenomena such as traffic flow. In this paper, we propose a novel inviscid, non-local regularisation in non-divergence form. The salient feature of our approach is that we can obtain sharp a priori estimates on the total variation and supremum norm, and justify the singular limit for Lipschitz initial data up to the time of catastrophe. For generic conservation laws, this result is sharp, since we can demonstrate non-convergence when the initial data features simple discontinuities. Conservation laws with linear flux derivative, such as the Burgers equation, behave better in the presence of discontinuities. Hence, we devote special attention to the limiting behaviour of non-local solutions with respect to the Burgers equation for a simple class of discontinuous initial data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_16743 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Non-Conservative, Non-Local Approximation of the Burgers Equation Ghoshal, Shyam Sundar Venkatesh, Parasuram Wiedemann, Emil Analysis of PDEs 35B44, 35B65, 35L65, 35L67 (Primary) 35D30, 35L03 (Secondary) The analysis of non-local regularisations of scalar conservation laws is an active research program. Applications of such equations are found in the modelling of physical phenomena such as traffic flow. In this paper, we propose a novel inviscid, non-local regularisation in non-divergence form. The salient feature of our approach is that we can obtain sharp a priori estimates on the total variation and supremum norm, and justify the singular limit for Lipschitz initial data up to the time of catastrophe. For generic conservation laws, this result is sharp, since we can demonstrate non-convergence when the initial data features simple discontinuities. Conservation laws with linear flux derivative, such as the Burgers equation, behave better in the presence of discontinuities. Hence, we devote special attention to the limiting behaviour of non-local solutions with respect to the Burgers equation for a simple class of discontinuous initial data. |
| title | A Non-Conservative, Non-Local Approximation of the Burgers Equation |
| topic | Analysis of PDEs 35B44, 35B65, 35L65, 35L67 (Primary) 35D30, 35L03 (Secondary) |
| url | https://arxiv.org/abs/2410.16743 |