Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider the shallow-water equations with Manning friction and topography source terms. The main purpose of this work concerns the derivation of a non-negativity preserving and ll-balanced scheme that approximates solutions of the system and preserves the associated steady states, including the moving ones. In addition, the scheme has to deal with vanishing water heights and transitions beten t and dry areas. To address such issues, a particular attention is paid to the study of the steady states related to the friction source term. Then, a Godunov-type scheme is obtained by using a relevant average of the source terms in order to enforce the required ll-balance property. An implicit treatment of both topography and friction source terms is also exhibited to improve the scheme while dealing with vanishing water heights. A second-order ll-balanced MUSCL extension is designed, as ll as an extension for the two-dimensional case. Numerical experiments are performed in order to highlight the properties of the scheme.
['Victor Michel-Dansac', 'Christophe Berthon', 'Stéphane Clain', 'Françoise Foucher']
I am writing a research paper on a idea A well-balanced scheme for the shallow-water equations with topography or Manning friction
[{"from": "human", "value": "I am writing a research paper on a idea A well-balanced scheme for the shallow-water equations with topography or Manning friction"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider the shallow-water equations with Manning friction and topography source terms. The main purpose of this work concerns the derivation of a non-negativity preserving and ll-balanced scheme that approximates solutions of the system and preserves the associated steady states, including the moving ones. In addition, the scheme has to deal with vanishing water heights and transitions beten t and dry areas. To address such issues, a particular attention is paid to the study of the steady states related to the friction source term. Then, a Godunov-type scheme is obtained by using a relevant average of the source terms in order to enforce the required ll-balance property. An implicit treatment of both topography and friction source terms is also exhibited to improve the scheme while dealing with vanishing water heights. A second-order ll-balanced MUSCL extension is designed, as ll as an extension for the two-dimensional case. Numerical experiments are performed in order to highlight the properties of the scheme."}]