Numerical simulation of conservation laws with moving grid nodes: Application to tsunami wave modelling

TL;DR

This paper presents a novel finite volume scheme on moving grids for simulating conservation laws, with applications to tsunami modeling, achieving high resolution without solution smearing and maintaining conservation properties.

Contribution

It introduces a dynamic grid movement technique combined with a predictor-corrector method, enhancing resolution in areas with large gradients without interpolation.

Findings

Effective in resolving large solution gradients

Maintains conservation and high accuracy

Applicable to complex shallow water wave simulations

Abstract

In the present article we describe a few simple and efficient finite volume type schemes on moving grids in one spatial dimension combined with appropriate predictor-corrector method to achieve higher resolution. The underlying finite volume scheme is conservative and it is accurate up to the second order in space. The main novelty consists in the motion of the grid. This new dynamic aspect can be used to resolve better the areas with large solution gradients or any other special features. No interpolation procedure is employed, thus unnecessary solution smearing is avoided, and therefore, our method enjoys excellent conservation properties. The resulting grid is completely redistributed according the choice of the so-called monitor function. Several more or less universal choices of the monitor function are provided. Finally, the performance of the proposed algorithm is illustrated on several examples stemming from the simple linear advection to the simulation of complex shallow water waves. The exact well-balanced property is proven. We believe that the techniques described in our paper can be beneficially used to model tsunami wave propagation and run-up.