Multi-scale time integration schemes for three-dimensional hydrodynamic marine models by Ange Pacifique ISHIMWE

The coastal ocean, covering a mere 8% of Earth's surface and 0.2% of the global ocean volume, plays a pivotal role in human activities and is thus crucial for study. These domains have to deal with large aspect ratio domains where the horizontal length scales can reach several thousands of kilometers while the depth is generally of the order of one kilometer or smaller. For this reason, numerical models of marine hydrodynamics must address a wide range of temporal and spatial scales.  To achieve both efficiency and numerical stability, careful selection of time-stepping schemes is necessary. One common approach is to separate fast and slow dynamics into distinct modes, with fast waves modeled using a two-dimensional system through depth averaging, while slower motions are treated in three dimensions.

The objective of this thesis is to implement a second-order split-time scheme for hydrodynamic equations to address this complex configuration. One of the main goal is to take into account the different computaion cost of each terms, thereby enhancing simulation efficiency. Another goal is to reduce numerical diffusion, making second-order methods preferable. To achieve this, a new split-explicit Runge-Kutta scheme is developed and adapted for the Discontinuous-Galerkin Finite Element method to establish a second-order time-stepping approach for more precise results. This method integrates a three-stage low-storage Runge-Kutta scheme for slow processes and a two-stage low-storage scheme for faster processes. However, while functional in theoretical scenarios, limitations arise from vertical dynamics, such as turbulent diffusivity or viscosity. To address this, an implicit component is incorporated into the temporal scheme. Both models undergo successful testing across various scenarios, demonstrating their order of convergence, ability to reduce numerical diffusion, and overall accuracy.

Jury members :

• Prof. Eric Deleersnijder (UCLouvain, Belgium), supervisor
• Prof. Vincent Legat (UCLouvain, Belgium), supervisor
• Prof. Paul Fisette (UCLouvain, Belgium), chairperson
• Prof. Emmanuel Hanert (UCLouvain, Belgium)
• Dr. Jonathan Lambrechts (UCLouvain, Belgium)
• Prof. Jochen Schuetz (UHasselt, Belgium)
• Prof. Jean-Marie Beckers (ULiège, Belgium)

Visio conference link : https://teams.microsoft.com/l/meetup-join/19%3ameeting_YzgzZjY2MTItZWMyZi00Y2Y4LWIwYmYtNmFhMzc5NzY1NmEx%40thread.v2/0?context=%7b%22Tid%22%3a%227ab090d4-fa2e-4ecf-bc7c-4127b4d582ec%22%2c%22Oid%22%3a%22ea3d55bf-b520-4806-ba2c-9bd5ca19898b%22%7d