7 research outputs found

    Implicit and semi-implicit well-balanced finite-volume methods for systems of balance laws

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    The aim of this work is to design implicit and semi-implicit high-order well-balanced finite-volume numerical methods for 1D systems of balance laws. The strategy introduced by two of the authors in some previous papers for explicit schemes based on the application of a well-balanced reconstruction operator is applied. The well-balanced property is preserved when quadrature formulas are used to approximate the averages and the integral of the source term in the cells. Concerning the time evolution, this technique is combined with a time discretization method of type RK-IMEX or RK-implicit. The methodology will be applied to several systems of balance laws.This work is partially supported by projects RTI2018-096064-B-C21 funded by MCIN/AEI/10.13039/501100011033 and “ERDF A way of making Europe”, projects P18-RT-3163 of Junta de Andalucía and UMA18-FEDERJA-161 of Junta de Andalucía-FEDER-University of Málaga. G.Russo and S.Boscarino acknowledge partial support from the Italian Ministry of University and Research (MIUR), PRIN Project 2017 (No. 2017KKJP4X) entitled “Innovative numerical methods for evolu-tionary partial differential equations and applications”. I. Gómez-Bueno is also supported by a Grant from “El Ministerio de Ciencia, Innovación y Universidades”, Spain (FPU2019/01541) funded by MCIN/AEI/10.13039/501100011033 and “ESF Invest-ing in your future”. // Funding for open access charge: Universidad de Málaga/CBUA

    Discontinuous Galerkin scheme for the spherical shallow water equations with applications to tsunami modeling and prediction

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    We present a novel high-order discontinuous Galerkin discretization for the spherical shallow water equations, able to handle wetting/drying and non-conforming, curved meshes in a well-balanced manner. This requires a well-balanced discretization, that cannot rely on exact quadrature, due to the curved mesh. Using the strong form of the discontinuous Galerkin discretization, we achieve a splitting of the well-balanced condition into individual problems for the flux and volume terms, which has significant advantages: It allows for the construction of non-conforming, well-balanced flux discretizations, i.e. we can perform non- conforming mesh refinement while preserving the well-balanced property of the scheme. More importantly, this approach enables the development of a new method for handling wet/dry transitions. In contrast to other wetting/drying methods, it is well-balanced and able to handle wetting/drying robustly at any polynomial order, without the introduction of physical model assumptions such as viscosity, artificial porosity or cancellation of gravity. We perform a series of one-dimensional tests and analyze the properties of our scheme. In order to validate our method for the simulation of large-scale tsunami events on the rotating sphere, we perform numerical simulations of the 2011 Tohoku tsunami and compare our results to real-world buoy data. The method is able to predict arrival times and wave amplitudes accurately even over long distances. This indicates that our method accurately captures all physical phenomena relevant to the long-term evolution of tsunami waves
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