20 research outputs found
On supraconvergence phenomenon for second order centered finite differences on non-uniform grids
In the present study we consider an example of a boundary value problem for a
simple second order ordinary differential equation, which may exhibit a
boundary layer phenomenon. We show that usual central finite differences, which
are second order accurate on a uniform grid, can be substantially upgraded to
the fourth order by a suitable choice of the underlying non-uniform grid. This
example is quite pedagogical and may give some ideas for more complex problems.Comment: 26 pages, 2 figures, 2 tables, 37 references. Other author's papers
can be downloaded at http://www.denys-dutykh.com
Numerical simulation of conservation laws with moving grid nodes: Application to tsunami wave modelling
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.Comment: 46 pages, 7 figures, 7 tables, 94 references. Accepted to
Geosciences. Other author's papers can be downloaded at
http://www.denys-dutykh.com
A new run-up algorithm based on local high-order analytic expansions
The practically important problem of the wave run-up is studied in this
article in the framework of Nonlinear Shallow Water Equations (NSWE). The main
novelty consists in the usage of high order local asymptotic analytical
solutions in the vicinity of the shoreline. Namely, we use the analytical
techniques introduced by S. Kovalevskaya and the analogy with the compressible
gas dynamics (i.e. gas outflow problem into the vacuum). Our run-up algorithm
covers all the possible cases of the wave slope on the shoreline and it
incorporates the new analytical information in order to determine the shoreline
motion to higher accuracy. The application of this algorithm is illustrated on
several important practical examples. Finally, the simulation results are
compared with the well-known analytical and experimental predictions.Comment: A minor update: 27 pages, 11 figures, 2 tables, 46 references. Other
author's papers can be viewed at http://www.denys-dutykh.com
Numerical Simulation of Conservation Laws with Moving Grid Nodes: Application to Tsunami Wave Modelling
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
Numerical modelling of surface water wave interaction with a moving wall
73 pages, 15 figures, 1 table, 80 references. Published in Commun. Comput. Phys. In this version minor typos were corrected. Other author's papers can be downloaded at http://www.denys-dutykh.com/International audienceIn the present manuscript, we consider the practical problem of wave interaction with a vertical wall. However, the novelty here consists in the fact that the wall can move horizontally due to a system of springs. The water wave evolution is described with the free surface potential flow model. Then, a semi-analytical numerical method is presented. It is based on a mapping technique and a finite difference scheme in the transformed domain. The idea is to pose the equations on a fixed domain. This method is thoroughly tested and validated in our study. By choosing specific values of spring parameters, this system can be used to damp (or in other words to extract the energy of) incident water waves
Long wave interaction with a partially immersed body. Part I: Mathematical models
Accepted to Comm. Comp. Phys.: 63 pages, 5 figures, 103 references, 1 appendix. Other author's papers can be downloaded at http://www.denys-dutykh.com/International audienceIn the present article we consider the problem of wave interaction with a partially immersed, but floating body. We assume that the motion of the body is prescribed. The general mathematical formulation for this problem is presented in the framework of a hierarchy of mathematical models. Namely, in this first part we formulate the problem at every hierarchical level. The special attention is payed to fully nonlinear and weakly dispersive models since they are most likely to be used in practice. For this model we have to consider separately the inner (under the body) and outer domains. Various approached to the gluing of solutions at the boundary is discussed as well. We propose several strategies which ensure the global conservation or continuity of some important physical quantities
Dispersive shallow water wave modelling. Part IV: Numerical simulation on a globally spherical geometry
51 pages, 14 figures, 1 table, 91 references. Other author's papers can be downloaded at http://www.denys-dutykh.com/International audienceIn the present manuscript we consider the problem of dispersive wave simulation on a rotating globally spherical geometry. In this Part IV we focus on numerical aspects while the model derivation was described in Part III. The algorithm we propose is based on the splitting approach. Namely, equations are decomposed on a uniform elliptic equation for the dispersive pressure component and a hyperbolic part of shallow water equations (on a sphere) with source terms. This algorithm is implemented as a two-step predictor-corrector scheme. On every step we solve separately elliptic and hyperbolic problems. Then, the performance of this algorithm is illustrated on model idealized situations with even bottom, where we estimate the influence of sphericity and rotation effects on dispersive wave propagation. The dispersive effects are quantified depending on the propagation distance over the sphere and on the linear extent of generation region. Finally, the numerical method is applied to a couple of real-world events. Namely, we undertake simulations of the Bulgarian 2007 and Chilean 2010 tsunamis. Whenever the data is available, our computational results are confronted with real measurements