Abdelaziz Beljadid

Abdelaziz Beljadid, Philippe G. LeFloch

We introduce a second-order, central-upwind finite volume method for the discretization of nonlinear hyperbolic conservation laws posed on the two-dimensional sphere. The semi-discrete version of the proposed method is based on a technique of local propagation speeds and it is free of any Riemann solver. The main advantages of our scheme are the high resolution of discontinuous solutions, its low numerical dissipation, and its simplicity for the implementation. The proposed scheme does not use any splitting approach, which is applied in some cases to upwind schemes in order to simplify the resolution of Riemann problems. The semi-discrete form of the scheme is strongly linked to the analytical properties of the nonlinear conservation law and to the geometry of the sphere. The curved geometry is treated here in an analytical way so that the semi-discrete form of the proposed scheme is consistent with a geometric compatibility property. Furthermore, the time evolution is carried out by using a total-variation-diminishing Runge-Kutta method. A rich family of (discontinuous) stationary solutions is available for the problem under consideration when the flux is nonlinear and foliated (as identified by the author in an earlier work). We present here a series of numerical examples, obtained by considering non-trivial steady state solutions and this leads us to a good validation of the accuracy and efficiency of the proposed central-upwind finite volume method. Our numerical tests confirm the stability of the proposed scheme and clearly show its ability to capture accurately discontinuous steady state solutions to nonlinear hyperbolic conservation laws posed on the sphere.

Abderrahmane Benfanich, Yves Bourgault, Abdelaziz Beljadid

Standard finite element discretizations of the Richards equation may violate the discrete minimum principle, producing unphysical negative saturations. While existing bound-preserving methods typically rely on computationally expensive fully implicit solvers, we propose a novel semi-implicit finite element framework utilizing a bounded continuous auxiliary variable. Our approach treats the gravity-driven advective term using a linearly implicit technique, which improves the time-step restrictions required by explicit gravity methods near the degenerate limit. We provide rigorous mathematical proofs establishing sufficient geometric and algebraic constraints for discrete positivity and the discrete maximum principle, specifically a local Péclet condition and a discrete row-sum condition. When both conditions are satisfied on weakly acute meshes with mass lumping, our framework ensures that numerical solutions strictly respect physical bounds across highly degenerate conditions and initially dry soil regimes. Comprehensive numerical validation demonstrates the method across multiple flow regimes, including cases where algebraic conditions are satisfied, violated, and recovered through mesh refinement.