Mehmet Ersoy

We study the three-dimensional compressible primitive equations under the hydrostatic approximation in an isothermal regime. Exploiting the hydrostatic balance, we introduce a stratification change of variables that removes the vertical dependence of the effective density and yields a reduced system coupling a two-dimensional transport equation for the density with an anisotropic parabolic equation for the horizontal velocity, supplemented by a reconstruction formula for the vertical velocity. We analyze the associated anisotropic viscous operator with periodic boundary conditions in the horizontal variables and homogeneous Neumann boundary conditions in the vertical direction, and establish its analytic semigroup property together with L p maximal regularity. In a Danchin-type functional framework based on real interpolation spaces, we prove local-in-time existence and uniqueness of strong solutions for strictly positive initial density and sufficiently regular initial velocity. The proof relies on transport estimates in W 1,r (T 2 ), maximal-regularity estimates for variable density coefficients of Solonnikov-Danchin type, and a fixed-point argument that closes the nonlinear coupling while preserving positivity and bounds of the density.

The choice of the mesh refinement threshold is a long-standing and problem-dependent issue in adaptive mesh refinement (AMR) methods, often constituting a critical weakness due to the need for ad hoc user calibration. In this work, we propose an automatic mesh refinement thresholding strategy designed to detect all relevant regions requiring refinement without any hand-tuned parameter. The approach is implemented within a block-based adaptive mesh refinement (BB-AMR) finite-volume framework for the multidimensional Saint-Venant system, enabling efficient mesh handling and straightforward parallelisation, while remaining compatible with other numerical discretisations. The automatic mesh refinement threshold (AMRT) is constructed from the decreasing rearrangement (distribution) function of the refinement indicator, enabling a robust identification of dynamically significant scales. This strategy provides a reliable balance between numerical accuracy and computational cost, while removing the dependence on user-defined thresholds. The resulting AMR method is robust, efficient, and nearly parameter-free. Its performance is assessed through a series of demanding numerical benchmarks related to tsunami propagation, including wave propagation over complex bathymetries and three-dimensional run-up configurations, where the method demonstrates both accuracy and significant computational savings.

We propose a one-dimensional Saint-Venant (open-channel) model for overland flows, including a water input–output source term modeling recharge via rainfall and infiltration (or exfiltration). We derive the model via asymptotic reduction from the two-dimensional Navier–Stokes equations under the shallow water assumption, with boundary conditions including recharge via ground infiltration and runoff. This new model recovers existing models as special cases, and adds more scope by adding water-mixing friction terms that depend on the rate of water recharge. We propose a novel entropy function and its flux, which are useful in validating the model’s conservation or dissipation properties. Based on this entropy function, we propose a finite volume scheme extending a class of kinetic schemes and provide numerical comparisons with respect to the newly introduced mixing friction coefficient. We also provide a comparison with experimental data.

We propose a one-dimensional Saint-Venant (open channel) model overland flows including a water input--output source term modelling recharge via rainfall and infiltration (or exfiltration). We derive the model via asymptotic reduction from the two-dimensional Navier--Stokes equations under the shallow water assumption, with boundary conditions including recharge via ground infiltration and runoff. This new model recovers existing models as sepcial cases, and adds more scope by adding a water-mixing friction terms that depends on the rate of water recharge. We prospose a novel entropy function and its flux, that are useful in validating the model's conservation or dissipation properties. Based on this entropy function we propose a finite volume scheme extending a class of kinetic schemes and provide numerical comparisons with respect to the newly introduced mixing friction coefficient. We also provide a comparison with experimental data.

We present a non-linear dispersive shallow water model which enters in the framework of section-averaged models. These new equations are derived up to the second order of the shallow water approximation starting from the three-dimensional incompressible and irrotational Euler system. The derivation is carried out in the case of non uniform rectangular section and it generalises the well-known one-dimensional Serre-Green-Naghdi (SGN) equations on uneven bottom. The obtained model is fully consistent with the Euler system. We propose a well-balanced finite volume approximation and we present some numerical results to show the influence of the section variation.

In this paper, we present a new non-linear dispersive model for open channel and river flows. These equations are the second order shallow water approximation of the section-averaged (three-dimensional) incompressible and irrotational Euler system. This new asymptotic model generalises the well-known one-dimensional Serre-Green-Naghdi (SGN) equations for rectangular section on uneven bottom to arbitrary chan-nel/river section.

Numerical simulations of multi dimensional large scale fluid-flows such as tsunamis, are still nowadays a challenging and a difficult problem. To this purpose, a parallel finite volume scheme on adaptive unstructured meshes for multi dimensional Saint-Venant system is presented. The adaptive mesh refinement method is based on a block-based decomposition (called BB-AMR) which allows quick meshing and easy parallelization. The main difficulty addressed here concerns the selection of the mesh refinement threshold which is certainly the most important parameter in the AMR method. Usually, the threshold is calibrated according to the test problem to balance the accuracy of the solution and the computational cost. To avoid " hand calibration " , we apply an automatic threshold method based on the decreasing rearrangement function of the mesh refinement criterion. This method is applied and validated successfully to the one and two dimensional non homogeneous Saint-Venant system through several tsunamis propagation test cases.

We present the full derivation of a one dimensional “Saint–Venant” like equations for barotropic compressible pipe flows including friction. The one dimensional hyperbolic system is called γ -pressurized model where γ is the adiabatic constant. It is