Research

The application domains of my research include atmospheric circulation modeling (using the shallow water equations), urban flood modeling (using the shallow water and porosity-based shallow water equations) and nearshore wave propagation modeling (using nonlinear dispersive models such as the KdV and Boussinesq equations). Numerical methods include finite differences, volumes, spectral methods, semi-Lagrangian methods, exponential methods and semi-Lagrangian exponential methods, as well as parallel-in-time and domain decomposition methods.

Predictor-corrector iterative Parallel-in-Time (PinT) methods, such as Parareal, MGRIT and PFASST, seek to replace the traditional serial time-stepping approach by the simultaneous computation of several time steps. Their objective is to take more advantage of massively parallel computing systems and accelerate the numerical simulation of time-dependent problems. However, despite their successful application to parabolic problems, PinT methods suffer from stability and convergence issues when applied to simple hyperbolic ones, which has discouraged their application to more complex advection-dominated models, such as those arising in fluid dynamics. My main objective is to better understand these issues, study parametric choices and formulate new PinT approaches in order to give steps towards the temporal parallelization of complex and operational fluid problems.

Domain decomposition methods (DDM) allow solving spatial-dependent differential equations by decomposing the spatial domain into two or more subdomains and solving the problem in each of them. The Schwarz methods, formulated by Schwarz in 1870 and further developed by Lions in the 1980s in the context of parallel computing, are iterative DDMs: when dividing the spatial domain, one needs to define unknown boundary conditions at the artificial interfaces, and the iterations allow correcting these conditions progressively. Among the several parameters determining the convergence of Schwarz methods, my research mainly focuses on formulating and studying optimal boundary operators, called transparent boundary conditions (TBCs), at the interface between subdomains, both on the continuous and discrete levels. These optimal operators yield immediate convergence of the DDM method, but are, in general, difficult to formulate and implement, even in the case of linear problems. The main applications of my work in this subject concern fluid dynamics problems.