Numerical Simulation of Conservation Laws with Moving Grid Nodes: Application to Tsunami Wave Modelling

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 an appropriate predictor–corrector method to achieve higher resolutions. 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 to 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.

Adaptive Numerical Modelling of Tsunami Wave Generation and Propagation with FreeFem++

“…We present the evidence here, following the work done for the 1D case of the BBM-BBM system in [33], that the second order RUNGE-KUTTA time scheme considered for the sBBM variable bottom in space is of order 2. We note that the function ζ(x, y, t) is only used for the generation of tsunami 5 .01/2 0 0.24145 -1.10773 -0.60317 -1.62575 -2 6 .01/2 1 0.06078 1.990 0.28016 1.983 0.30196 0.998 0.81276 1.000 2 7 .01/2 2 0.01524 1.996 0.07038 1.993 0.15119 0.998 0.40696 0.999 2 8 .01/2 3 0.00381 1.999 0.01760 1.999 0.07578 0.998 0.20355 0.999 wave and, thus, will not be taken into account in the convergence rate test. In this example, we take bi-periodic Boundary Conditions (BC) for η h , u h and v h on the whole boundary of the square [0, 2L] × [0, 2L], where L = 50 and we consider the following exact solutions:…”

“…The FreeFem++ programming framework offers the advantage to hide all technical issues related to the implementation of the finite element method. Traditionally, tsunami waves are modelled using hydrostatic models [4][5][6][7]. In the present manuscript we employ a non-hydrostatic BOUSSINESQ-type system to be specified below.…”

Numerical Simulation of Feller’s Diffusion Equation

“…Working with the pseudo-inverse allows to overcome the issue of the retention phenomenon, which manifests as the expanding support of p (x, t) for positive (and possibly large) times t > 0 , t ≫ 1 , since the computational domain was transformed to [ 0, 1 ] . This method is the Lagrangian counterpart of the moving mesh technique in the Eulerian setting [15,16]. A simple Matlab code, which implements the scheme we described hereinabove, is freely available for reader's convenience at this URL address: https://github.com/dutykh/Feller/ * In our code we employed the simplest forward finite differences and it lead satisfactory results.…”

Staggered Conservative Scheme for 2-Dimensional Shallow Water Flows

“…Because analytical solutions of the SWE are only available in very few simple and ideal situations, it is important to develop robust and accurate numerical methods to solve SWE in more realistic engineering applications. Extensive research on numerical models for SWE has been developed for a long time, and can generally be classified into three types, i.e., the finite element method [1], the finite difference method [2], or the finite volume method [3].…”