High-order well-balanced numerical schemes for shallow-water systems with Coriolis terms
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The goal of this work is to develop LDG high-order well-balanced schemes for the shallow-water equations with Coriolis terms. We will follow the strategy presented in \cite{bib:castro-Lopez-Pares-08}-\cite{bib:Castro-Pares-20} for building well-balanced methods by considering well-balanced reconstruction operators. The critical step of this procedure is solving the a stationary PDE problem for each cell and time step. In this work we combine local and global solvers to determine stationary solutions for the system. Once the local solution is obtained at the volume, it is extended to the whole considered stencil. We focus in the case where it is not possible to obtain the exact solutions the stationary system. In this case, the exact stationary solution is replaced by a suitable approximation. Following \cite{igomez2021}, in this case, the semi-discrete finite volume numerical scheme is well-balanced, if the sequence of cell averages computed from the approximation is an equilibrium of the system of the ODE system given by the semidiscrete scheme. In this work we propose techniques based in nonlinear optimization and in Deep Learning to compute the stationary discrete solutions with the desired properties.
