60 research outputs found
On the Galilean invariance of some dispersive wave equations
Surface water waves in ideal fluids have been typically modeled by asymptotic
approximations of the full Euler equations. Some of these simplified models
lose relevant properties of the full water wave problem. One of them is the
Galilean symmetry, which is not present in important models such as the BBM
equation and the Peregrine (Classical Boussinesq) system. In this paper we
propose a mechanism to modify the above mentioned classical models and derive
new, Galilean invariant models. We present some properties of the new
equations, with special emphasis on the computation and interaction of their
solitary-wave solutions. The comparison with full Euler solutions shows the
relevance of the preservation of Galilean invariance for the description of
water waves.Comment: 29 pages, 13 figures, 2 tables, 71 references. Other author papers
can be downloaded at http://www.denys-dutykh.com
Generation of two-dimensional water waves by moving bottom disturbances
We investigate the potential and limitations of the wave generation by
disturbances moving at the bottom. More precisely, we assume that the wavemaker
is composed of an underwater object of a given shape which can be displaced
according to a prescribed trajectory. We address the practical question of
computing the wavemaker shape and trajectory generating a wave with prescribed
characteristics. For the sake of simplicity we model the hydrodynamics by a
generalized forced Benjamin-Bona-Mahony (BBM) equation. This practical problem
is reformulated as a constrained nonlinear optimization problem. Additional
constraints are imposed in order to fulfill various practical design
requirements. Finally, we present some numerical results in order to
demonstrate the feasibility and performance of the proposed methodology.Comment: 21 pages, 7 figures, 1 table, 69 references. Other author's papers
can be downloaded at http://www.denys-dutykh.com
Theoretical and computational structures on solitary wave solutions of Benjamin Bona Mahony-Burgers equation
This paper aims to obtain exact and numerical solutions of the nonlinear Benjamin Bona
Mahony-Burgers (BBM-Burgers) equation. Here, we propose the modi ed Kudryashov method for getting the exact traveling wave solutions of BBM-Burgers equation and a septic B-spline collocation nite element method for numerical investigations. The numerical method is validated by studying solitary wave motion. Linear stability analysis of the numerical scheme
is done with Fourier method based on von-Neumann theory. To show suitability and robustness of the new numerical algorithm, error norms L2, L1 and three invariants I1; I2 and I3 are calculated and obtained results are given both numerically and graphically. The obtained results state that our exact and numerical schemes ensure evident and they are penetrative mathematical instruments for solving nonlinear evolution equation
The Lie-group method based on radial basis functions for solving nonlinear high dimensional generalized BenjaminâBonaâMahonyâBurgers equation in arbitrary domains
The aim of this paper is to introduce a new numerical method for solving the nonlinear generalized BenjaminâBonaâMahonyâBurgers (GBBMB) equation. This method is combination of group preserving scheme (GPS) with radial basis functions (RBFs), which takes advantage of two powerful methods, one as geometric numerical integration method and the other meshless method. Thus, we introduce this method as the Lie-group method based on radial basis functions (LGâRBFs). In this method, we use Kansas approach to approximate the spatial derivatives and then we apply GPS method to approximate first-order time derivative. One of the important advantages of the developed method is that it can be applied to problems on arbitrary geometry with high dimensions. To demonstrate this point, we solve nonlinear GBBMB equation on various geometric domains in one, two and three dimension spaces. The results of numerical experiments are compared with analytical solutions and the method presented in Dehghan et al. (2014) to confirm the accuracy and efficiency of the presented method
Artificial boundary conditions for the linearized Benjamin-Bona-Mahony equation
International audienceWe consider various approximations of artificial boundary conditions for linearized Benjamin-Bona-Mahoney equation. Continuous (respectively discrete) artificial boundary conditions involve non local operators in time which in turn requires to compute time convolutions and invert the Laplace transform of an analytic function (respectively the Z-transform of an holomorphic function). In this paper, we derive explicit transparent boundary conditions both continuous and discrete for the linearized BBM equation. The equation is discretized with the Crank Nicolson time discretization scheme and we focus on the difference between the upwind and the centered discretization of the convection term. We use these boundary conditions to compute solutions with compact support in the computational domain and also in the case of an incoming plane wave which is an exact solution of the linearized BBM equation. We prove consistency, stability and convergence of the numerical scheme and provide many numerical experiments to show the efficiency of our tranparent boundary conditions
Galerkin finite element solution for benjamin-bona-mahony-burgers equation with cubic b-splines
In this article, we study solitary-wave solutions of the nonlinear BenjaminâBonaâMahonyâ
Burgers(BBMâBurgers) equation based on a lumped Galerkin technique using cubic Bspline finite elements for the spatial approximation. The existence and uniqueness of
solutions of the Galerkin version of the solutions have been established. An accuracy
analysis of the Galerkin finite element scheme for the spatial approximation has been well
studied. The proposed scheme is carried out for four test problems including dispersion
of single solitary wave, interaction of two, three solitary waves and development of an
undular bore. Then we propose a full discrete scheme for the resulting IVP. Von Neumann
theory is used to establish stability analysis of the full discrete numerical algorithm. To
display applicability and durableness of the new scheme, error norms L2, Lâ and three
invariants I1, I2 and I3 are computed and the acquired results are demonstrated both
numerically and graphically. The obtained results specify that our new scheme ensures
an apparent and an operative mathematical instrument for solving nonlinear evolution
equation
Numerical Analysis of a Linear-Implicit Average Scheme for Generalized Benjamin-Bona-Mahony-Burgers Equation
A linear-implicit finite difference scheme is given for the initial-boundary problem of GBBMBurgers equation, which is convergent and unconditionally stable. The unique solvability of numerical solutions is shown. A priori estimate and second-order convergence of the finite difference approximate solution are discussed using energy method. Numerical results demonstrate that the scheme is efficient and accurate
Finite volume methods for unidirectional dispersive wave models
We extend the framework of the finite volume method to dispersive
unidirectional water wave propagation in one space dimension. In particular we
consider a KdV-BBM type equation. Explicit and IMEX Runge-Kutta type methods
are used for time discretizations. The fully discrete schemes are validated by
direct comparisons to analytic solutions. Invariants conservation properties
are also studied. Main applications include important nonlinear phenomena such
as dispersive shock wave formation, solitary waves and their various
interactions.Comment: 25 pages, 12 figures, 51 references. Other authors papers can be
downloaded at http://www.lama.univ-savoie.fr/~dutykh
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