Natural hazards like earthquakes, volcanoes, landslides, and tsunamis pose unpredictable risks with
significant social and economic impacts. They are unpredictable and the submarine environment
makes direct measurements extremely difficult. In this context, numerical simulations constrained
with high quality geological data provide the unique tool to propose efficient risk reduction strategies.
Disasters, like the Sumatra tsunami in 2004, the Tohoku tsunami in 2011, the Anak Krakatau volcano
tsunami in 2018 or even the very recent Hunga tsunami, are reminders that such phenomena still
need to be studied to reinforce our risk reduction strategies by innovative predictive models.
The goal of this project is the design and implementation of numerical schemes that will deal efficient
with the different spatial scales during the propagation and inundation phases of the tsunamis.
In order to numerically solve the mathematical equations that characterize free surface flows , a
mesh which subdivides the domain into a conformal tesselation is often used. The distribution and
shape of the elements which comprise the mesh can be adapted based on various methods to
distribute computational resources such that areas of higher complexity are given more resolution.
Static meshes, despite being refined a-priori around the coastline, are inherently not efficient for
tsunami simulations where an impulsive wave is initially concentrated in a narrow region and then
propagates over a certain distance. Mesh adaptation techniques have proved their efficiency in
improving numerical accuracy while reducing the overall computational cost in many scientific
domains. Several adaptive strategies have already been deployed for shallow water flows, including
tsunami applications: hierarchical mesh refinement [3,4], unstructured re-meshing [3,5] and r-
adaptation [6].
Resently, we worked on metric-based anisotropic mesh adaptation for the non-linear shallow
water equations in the presence of wet-dry fronts [2], demostrating its suitability for predicting coastal
run-up from incident waves. We used the hydrostatic code UHAINA [1] coupled with the MMG
remeshing software, and developed a tailored error indicator and a simple wet-dry interface
treatment for solution transfer.
We now aim to extend this work to dispersive coastal flows. The dispersive regularization of the
SWE results in more regular solutions, for which classic error models may be limited. A key point of
the project will be to propose error estimators well suited for non-linear dispersive wave propagation.
To our best knowledge, such an error model would be completely novel. In particular, local
smoothness of the solution will first be considered, then we will study phase errors and ways to
control them. We have to highlight that, up to the authors’ knowledge, mesh adaptation for dispersive
wave propagation is, up to now, only limited on structured meshes [4,7] and this will be the first time
unstructured meshes will be used. Once the adaptive process has been validated on academic test
cases, we will run large scale simulations of tsunamis in a realistic setting.
[1] Filippini, A.G., et al., 2018, UHAINA, XVèmes Journés Nationales Génie Côtier - Génie Civil,
DOI:10.5150/jngcgc.2018.006.
[2] Yuan, D., Kazolea, M., Barral, N., Ricchiuto, M., Anisotropic mesh adaptation for the non-linear shallow
water equations, https://inria.hal.science/hal-05614066v1.
[3]- Wallwork, J. G., Barral, N., Kramer, S., Ham, D., Piggott, M., 2020. Goal-oriented error estimation and mesh
adaptation for shallow water modelling. SN Applied Sciences, Springer Verlag, 2-6.
[4]- Berger, M.J., & LeVeque, R. J.,Implicit adaptive mesh refinement for dispersive tsunami propagation,
https://arxiv.org/pdf/2307.05816
[5] Blaise, S. & St-Cyr. A., A dynamic hp-adaptive discontinuous Galerkin method for shallow water flows on the
sphere with application to a global tsunami simulation. Mon. Weather Rev., 140(3):978-996, 2012.
[6] - Arpaia, L., & Ricchiuto, M., 2020. Well balanced residual distribution for the ALE spherical shallow water
equations on moving adaptive meshes. J. Comput. Phys., 405, 109173.
[7] Popinet, S., 2015. A quadtree-adaptive multigrid solver for the Serre–Green–Naghdi equations. J Comput.
Phys., 302,336-358
