# A new pseudo-spectral solver for 2D tsunami equations subjected to dynamic earthquake ground motion

Thomas Melkior, Harsha S. Bhat, Faisal Amlani

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Thomas Melkior, Harsha S. Bhat, Faisal Amlani. A new pseudo-spectral solver for 2D tsunami equations subjected to dynamic earthquake ground motion. 10th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COMPDYN 2025), Jun 2025, Rhodes Island, Greece. hal-05146410

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Submitted on 22 Jul 2025

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# A new pseudo-spectral solver for 2D tsunami equations subjected to dynamic earthquake ground motion

Thomas Melkior<fnref n="1" />, <fnref n="2" />*, Harsha S. Bhat<fnref n="2" />, Faisal Amlani<fnref n="1" />

## ABSTRACT

Although a lot of progress has been made on tsunami modelling, some historical tsunamis are still not fully understood (see [1] and references therein). In the case of seismogenic tsunamis, earthquake or rupture dynamics (and how they relate to tsunami generation) are often overlooked since, conventionally, earthquake perturbation is modelled by an input step representing the final ground displacement. The present work therefore aims at studying the effects (and signatures) of earthquake dynamics on tsunamigenesis, the complexity of which derives from the different scales involved: high-frequency seismic waves propagating over long distances (from the epicenter to the sea floor) and the corresponding tsunami waves with their lower frequencies and lower propagation speeds.

In order to address such numerically challenging configurations, this contribution discusses a new tsunami solver that takes advantage of a spectral Fourier continuation (FC) method [2] that is high order and avoids numerical dispersion (or “pollution”) errors. The particular tsunami equations considered here are the classical nonlinear shallow water wave equations modified to include time-dependent sea floor variations (i.e., dynamic ground displacement and velocity). The (numerically) dispersionless feature of this method is very valuable for long time or long distance propagation since errors do not accumulate over space and time, unlike classical (possibly high-order) finite difference or finite element-based solvers that often require considerable effort (in terms of computational cost and refinement) to mitigate their well-known dispersion errors.

This work discusses extensions to a proof-of-concept contribution that has invoked successfully the aforementioned spectral method on a 1D slice of Palu bay (Indonesia) to model the unusual tsunami that followed the strike-slip Sulawesi earthquake. Here, a complete 2D solver is presented to model tsunami generation and propagation by dynamic earthquake sources in higher spatial dimensions. Performance studies are presented attesting to high-order convergence and high efficiency, including comparisons with high-order finite differences. Validations through classical benchmark cases [3],[4] (including those employing measured data) are also presented to demonstrate the relevance of the solver for realistic settings and potential hazard assessment.

## References

[1] Amlani, et al., “Supershear shock front contribution to the tsunami from the 2018 Mw 7.5 Palu, Indonesia earthquake” Geophysical Journal International, vol.230 (2022)

[2] F. Amlani and O. P. Bruno, “An FC-based spectral solver for elastodynamic problems in general three-dimensional domains”, Journal of Computational Physics, vol. 307 (2016)

[3] A.Delis and T. Katsaounis, “Numerical solution of the two-dimensional shallow water equations by the application of relaxation methods”, Applied Mathematical Modelling, vol. 29 (2005)

[4] J. L. Hammack, “A note on tsunamis: their generation and propagation in an ocean of uniform depth,” Journal of Fluid Mechanics, vol. 60 (1973)

<footnote n="1">¹Université Paris-Saclay, CentraleSupélec, ENS Paris-Saclay, CNRS, LMPS - Laboratoire de Mécanique Paris-Saclay
91190, Gif-sur-Yvette, France
thomas.melkior@ens-paris-saclay.fr, faisal.amlani@ens-paris-saclay.fr</footnote>

<footnote n="2">²Laboratoire de Géologie, Ecole Normale Supérieure, CNRS-UMR 8538, PSL Research University
75006 Paris, France
harhasbhat@gmail.com</footnote>