Maurizio Tavelli

Maurizio Tavelli, Olindo Zanotti

In this paper we combine a flexible covariant formulation of the shallow water equations with the semi-implicit numerical scheme developed over the years by Casulli and collaborators. After adopting an orthogonal, but non-orthonormal, coordinate basis on two dimensional manifolds, and by writing the divergence of symmetric tensors in a way that avoids the introduction of Christoffel symbols, the shallow water equations preserve a very close resemblance to the usual one expressed in Cartesian coordinates. In this way, a stable semi-implicit scheme can be derived by using an implicit discretization for the gradient of surface elevation in the momentum equations and for the velocity in the continuity equation, with stability properties that are independent of the celerity. We have tested the new method over a variety of challenging benchmarks, including, among the others, the smooth wave propagation over a water globe and the deformation of an artery branch. Two appealing additional features make the method particularly powerful with respect to oceanographic applications: firstly, thanks to the wetting and drying ability of our semi-implicit approach, no pathological behaviors occur at the poles; secondly, the scheme is naturally well-balanced, and it is able to preserve perfect stationarity, up to machined precision, of the entire ocean configuration of the earth.

Michael Dumbser, Andrea Thomann, Maurizio Tavelli et al.

We introduce a novel structure-preserving vertex-staggered semi-implicit four-split discretization of a unified first order hyperbolic formulation of continuum mechanics that is able to describe at the same time fluid and solid materials within the same mathematical model. The governing PDE system goes back to pioneering work of Godunov, Romenski, Peshkov and collaborators. Previous structure-preserving discretizations of this system allowed to respect the curl-free properties of the distortion field and the specific thermal impulse in the absence of source terms and were consistent with the low Mach number limit with respect to the adiabatic sound speed. However, the evolution of the thermal impulse and the distortion field were still discretized explicitly, thus requiring a rather severe CFL stability restriction on the time step based on the shear sound speed and the finite, but potentially large, speed of heat waves. Instead, the new four-split semi-implicit scheme presented in this paper has a material time step restriction only. For this purpose, the governing PDE system is split into four subsystems: i) a convective subsystem, which is the only one that is treated explicitly; ii) a heat subsystem, iii) a subsystem containing momentum, distortion field and specific thermal impulse; iv) a pressure subsystem. The three subsystems ii)-iv) are all discretized implicitly, hence a rather mild CFL restriction based on the velocity of the continuum is imposed. The method is asymptotically consistent with the low Mach number limit and the stiff relaxation limits. Moreover, it maintains an exactly curl-free distortion field and thermal impulse in the case of linear source terms or in their absence. The scheme is benchmarked against classical test cases verifying its theoretical properties.

Elena Gaburro, Matej Klima, Mauro Bonafini et al.

This work investigates a novel approach for the high order evolution of hyperbolic PDEs using ADER discontinuous Galerkin schemes within a direct Arbitrary-Lagrangian-Eulerian (ALE) framework on 3D moving polyhedral meshes with topology changes. Our direct ALE method is based on the PDE integration over 4D (3D+time) space-time control volumes connecting the elements of two subsequent tessellations, so to simultaneously evolve the solution both in time and between the two different meshes in an effective and high order manner. In this way, we also avoid any complex and expensive projection-reconstruction techniques and any mesh intersection operation typical of indirect ALE schemes. The crucial step consists in the strategy for building space-time control volumes that also connect elements with different shapes and neighborhoods due to a change in topology. In fact, simply linking existing elements by collapsing or expanding their edges would leave a "hole" in the space-time domain. To fill it, we introduce additional degenerate elements that we call hole-like elements. These are 4D objects with zero 3D volume at both the beginning and end of the timestep, but which possess a strictly non-zero 4D space-time volume. Given the uniqueness of this space-time approach in 3D+time and the necessity of characterizing the geometry of such elements, the main objective of this paper is the formal geometrical and numerical description of the method as well as the presentation of new and intuitive visualization strategies. In particular, we provide a detailed characterization of the hole-like elements arising in correspondence to 2-3, 3-2, and 4-4 flips on the underlying Delaunay tetrahedralization. Finally, we numerically show that the method is fully conservative, satisfies the GCL and maintains the correct order of convergence even in the presence of frequent topology changes.