A Well-Balanced SPH-ALE Scheme for Shallow Water Applications

In this work, a new discretization of the source term of the shallow water equations with non-flat bottom geometry is proposed to obtain a well-balanced scheme. A Smoothed Particle Hydrodynamics Arbitrary Lagrangian-Eulerian formulation based on Riemann solvers is presented to solve the SWE. Moving-Least Squares approximations are used to compute high-order reconstructions of the numerical fluxes and, stability is achieved using the a posteriori MOOD paradigm. Several benchmark 1D and 2D numerical problems are considered to test and validate the properties and behavior of the presented schemes.

Non-overlapping Schwarz algorithms for the Incompressible Navier-Stokes equations with DDFV discretizations

We propose and analyze non-overlapping Schwarz algorithms for the domain decomposition of the unsteady incompressible Navier-Stokes problem with Discrete Duality Finite Volume discretizations. The design of suitable transmission conditions for the velocity and the pressure is a crucial issue. We establish the well-posedness of the method and the convergence of the iterative process, pointing out how the numerical fluxes influence the asymptotic problem which is intended to be a discretization of the Navier-Stokes equations on the entire computational domain.Finally we numerically illustrate the behavior and performances of the algorithm. We discuss on numerical grounds the impact of the parameters for several mesh geometries and we perform simulations of the flow past an obstacle with several domains.

Numerical comparison between simplified mathematical models for rock-ice avalanches

<p>Rock-ice avalanches correspond to three-phase mixtures composed of a liquid and of particles of rock and ice. The presence of ice inside the mixture plays a key role in the mobility of rock-ice avalanches, since the heat produced by basal friction and particle collisions induces its transformation into water. Due to this continuous supply of liquid to the mixture, rock-ice avalanches can threaten populations living in cold mountainous areas. Thus, for a good hazard assessment and management, there is the need to construct mathematical models able to predict the flow of rock-ice avalanches. In the literature, there exist only few models that deal with this type of mass flows (Pudasaini & Krautblatter 2014, Bartelt et al. 2018, Sansone et al. 2021). As proposed in Sansone et al. (2021), a framework of different simplified rock-ice avalanche models can be derived by starting from a complete three-phase approach and by imposing two specific assumptions, namely the isokinetic and incompressibility hypotheses. In this way, five classes of simplified approaches can be detected, and these mathematical models are characterized by different levels of approximations of the physics of rock-ice avalanches.</p><p>In this work, we provide some numerical solutions for the depth-integrated one-dimensional versions of all the simplified mathematical models detected in Sansone et al. (2021). These numerical solutions are constructed using three different numerical schemes that distinguish themselves from the way the numerical fluxes are evaluated. While one of the three chosen numerical methods evaluates the numerical fluxes without considering the eigenstructure of the systems of equations, the other two schemes take partially or entirely account of the eigenstructure of the equation systems. Due to the possible loss of hyperbolicity detectable in some simplified models, we consider as test cases the problems of the small perturbations of the flow depth and of the concentrations.</p><p>The first result of the analysis computed corresponds to the comparison between the numerical solutions derived from the three numerical schemes for each class of models. In this way, the responses of the different numerical methods to each equation system can be investigated. The second result consists in comparing numerically the different classes of simplified models detected by Sansone et al. (2021), thus allowing us to quantify the effects of the assumptions of each class of models on the flow dynamics.</p><p> </p><p><strong>References:</strong></p><p>Bartelt P., Christen M., Bühler Y., Buser O. (2018), <em>Thermomechanical modelling of rock avalanches with debris, ice and snow entrainment</em>. In 9th European Conference on Numerical Methods in Geotechnical Engineering (NUMGE), University of Porto, Porto, PORTUGAL.</p><p>Pudasaini S., Krautblatter M. (2014), <em>A two-phase mechanical model for rock-ice avalanches</em>. Journal of Geophysical Research: Earth Surface 119 (10), 2272-2290.</p><p>Sansone S., Rosatti G., Zugliani D. (2021), <em>A mathematical framework for modelling rock-ice avalanches</em>. Paper under review.</p>

Numerical approximation of the shallow water equations with coriolis source term

We investigate in this work a class of numerical schemes dedicated to the non-linear Shallow Water equations with topography and Coriolis force. The proposed algorithms rely on Finite Volume approximations formulated on collocated and staggered meshes, involving appropriate diffusion terms in the numerical fluxes, expressed as discrete versions of the linear geostrophic balance. It follows that, contrary to standard Finite-Volume approaches, the linear versions of the proposed schemes provide a relevant approximation of the geostrophic equilibrium. We also show that the resulting methods ensure semi-discrete energy estimates. Numerical experiments exhibit the efficiency of the approach in the presence of Coriolis force close to the geostrophic balance, especially at low Froude number regimes.