# High order ImEx method for the shallow water model

Maria Kazolea, Ralph Lteif, Martin Parisot

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Maria Kazolea, Ralph Lteif, Martin Parisot. High order ImEx method for the shallow water model. HONOM 2024 - High-Order NOnlinear numerical Methods for evolutionary PDEs: theory and applications., Sep 2024, Chania Crete, Greece. hal-04884370

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https://inria.hal.science/hal-04884370v1

Submitted on 13 Jan 2025

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Distributed under a Creative Commons Attribution 4.0 International License
High order ImEx method for the shallow water model

M. Kazolea†, R. Lteif† and M. Parisot†

† Team CARDAMOM, INRIA Bordeaux, France
(maria.kazolea@inria.fr, ralph.lteif@inria.fr, martin.parisot@inria.fr)

**Keywords:** shallow water equations, implicit explicit schemes, high order in time, low Froude number.

ABSTRACT

This poster is devoted to the development and analysis of a robust and efficient high order numerical scheme for the shallow water flows in the low Froude number limit. We focus on ocean and coastal simulations at different scales, in particular, on the variation of the Froude number that goes from 1 at the coastline to two or three orders less offshore. Due to the great ocean's depth, classical hyperbolic schemes like Riemann solvers are not efficient [1]. In order to propose an efficient method in such regime, a part of the system has to be considered implicitly, leading to an ImEx (Implicit Explicit) scheme. In order to limit the size and number of linear systems to be solved, the CPR scheme [2] is a good first order candidate. The CPR approach is a fully diagonal segregated method which only relies on the implicit treatment of the water height and hybrid mass fluxes using explicit velocities. The method allows to avoid resolution of large linear systems. Concerning the high order in time integration, several Runge-Kutta schemes can be found in the literature [3] in the context of ImEx schemes, however to limit the number of linear systems to solve, we focus on Cranck Nicolson schemes. For the space discretization, a classical second order MUSCL reconstruction is used. We finally show, thanks to one- and two-dimensional test cases, that the developed scheme achieves the theoretical second-order rate of convergence. Furthermore, we conduct a comparative analysis of CPU times between the ImEx and explicit schemes, revealing important computational savings with the ImEx scheme particularly under the low Froude regime.

References

[1] H. Guillard and C. Viozat, "On the behaviour of upwind schemes in the low mach number limit", *Computers & Fluids*, 28(1):63-86, 1999.

[2] M. Parisot and J.-P. Vila, "Centered potential regularization for the advection upstream splitting method", *SIAM Journal on Numerical Analysis*, 54(5):3083-3104, 2016.

[3] G. Dimarco, R. Loubère, V. Michel-Dansac, and M.-H. Vignal, "Second-order implicit-explicit total variation diminishing schemes for the Euler system in the low Mach regime", *Journal of Computational Physics*, 372:178-201, 2018.

The authors acknowledge the financial support of the ANR project LAGOON whose scope is to improve physical models and to increase the computational performance for storm-surge predictions.