Location
MC 5501
Speaker
Sacha Cardonna, Ph.D. candidate in Mathematics, Institut Montpelliérain Alexander Grothendieck − Montpellier, France.
Title
A high-order DG/FV convex property preserving scheme for hyperbolic systems with applications to shallow water flows and fluid-structure interaction
Abstract
We first introduce a monolithic DG–FV subcell convex property preserving framework for general systems of hyperbolic conservation laws. The method relies on an arbitrary high-order Discontinuous Galerkin (DG) discretization that is locally reinterpreted as a Finite-Volume (FV) scheme on a suitable sub-partition of each element. High-order fluxes are reconstructed at the subcell level, allowing us to recover a fully conservative FV-like formulation while retaining the accuracy and flexibility of the underlying DG approximation.
A central ingredient of the approach is the construction of blended numerical fluxes, defined as convex combinations of the reconstructed high-order fluxes and robust first-order FV fluxes. The blending coefficients are computed locally, at each time step and for each subcell interface, in order to enforce key convex constraints such as positivity, discrete maximum principles, or even entropy stability. The resulting scheme preserves conservation, maintains high-order accuracy in smooth regions, and automatically reduces to a stable low-order method near critical areas, where Gibbs phenomena could otherwise lead to numerical instabilities and significant degradation of the solution quality.
In a second part, we apply the framework to the nonlinear shallow water equations, where additional structural properties such as preservation of water-height positivity and well-balancing for motionless steady states are enforced at both the cell and subcell levels. If the time allows, we will show that the methodology can be also embedded into a very recent fluid–structure interaction model, where the hyperbolic PDEs are coupled with elliptic equations and some ODEs, illustrating that the proposed method can be integrated into strongly coupled and geometrically constrained systems while retaining robustness and high-order accuracy.