A well-balanced positivity-preserving discontinuous Galerkin method for shallow water models with variable density
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911518842421248 |
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| author | She, Jun Dong, Haiyun Li, Maojun Ma, Jianjun |
| author_facet | She, Jun Dong, Haiyun Li, Maojun Ma, Jianjun |
| contents | In this paper, we present a numerical scheme designed for coupled systems of variable-topography shallow water flow and solute transport. By integrating a variable-density system with an expression for relative density of mixtures, a novel formulation of the coupled system is derived. To ensure the well-balanced property, auxiliary variables are introduced to reformulate the variable-density shallow water equations into a new form, which is then discretized using the discontinuous Galerkin (DG) method with the Lax-Friedrichs (LF) flux as the numerical flux. By selecting appropriate values for the auxiliary variables, we demonstrate that the proposed method accurately preserves steady-state solutions under still water conditions, thereby verifying its well-balanced nature. Furthermore, sufficient conditions for preserving the positivity of both water depth and concentration are proposed and rigorously proven. A positivity-preserving limiter is introduced to enforce these conditions. Finally, a series of numerical examples are conducted to validate the computational accuracy and effectiveness of the proposed method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_14954 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A well-balanced positivity-preserving discontinuous Galerkin method for shallow water models with variable density She, Jun Dong, Haiyun Li, Maojun Ma, Jianjun Numerical Analysis In this paper, we present a numerical scheme designed for coupled systems of variable-topography shallow water flow and solute transport. By integrating a variable-density system with an expression for relative density of mixtures, a novel formulation of the coupled system is derived. To ensure the well-balanced property, auxiliary variables are introduced to reformulate the variable-density shallow water equations into a new form, which is then discretized using the discontinuous Galerkin (DG) method with the Lax-Friedrichs (LF) flux as the numerical flux. By selecting appropriate values for the auxiliary variables, we demonstrate that the proposed method accurately preserves steady-state solutions under still water conditions, thereby verifying its well-balanced nature. Furthermore, sufficient conditions for preserving the positivity of both water depth and concentration are proposed and rigorously proven. A positivity-preserving limiter is introduced to enforce these conditions. Finally, a series of numerical examples are conducted to validate the computational accuracy and effectiveness of the proposed method. |
| title | A well-balanced positivity-preserving discontinuous Galerkin method for shallow water models with variable density |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2603.14954 |