On the numerical modelling of the dynamics of weakly nonlinear waves with dispersion

This work describes one approach to the construction of finite difference methods for weakly nonlinear equations with dispersion. Its idea is that the properties of the difference schemes and the differential models which are approximated by them must be consistent. Our research consists of two stages. At the first stage we study the properties of nonlinearly dispersive models that are used as a basis for constructing computational algorithms. At the second stage we analyse the finite difference schemes whose dispersive properties are consistent with those of the approximated models. This work deals with the development of numerical methods for solving nonlinearly dispersive equations that describe the propagation of long water waves. In spite of the considerable progress in the numerical modelling of the dynamics of long waves (see, e.g. [5,9,11]), we encounter difficulties when constructing and justifying the numerical methods for solving these equations. This is mainly due to the fact that the nonlinearly dispersive models that are used in practice are poorly known. In particular, the question of conservation laws for most of the models remains open. The aim of the work is to describe one approach to the construction of finite difference methods for solving weakly nonlinear equations with dispersion. The idea of the approach is to ensure the efficiency of the computational methods by fitting the properties of difference schemes to those of the differential models approximated by them. This approach, as applied to nonlinear hyperbolic equations in gas dynamics, has yielded good results [8,10]. As to the numerical methods for solving nonlinearly dispersive equations, there are not many adequate results to date. Even the dissipation and dispersion of difference schemes (the terminology from [10]) for nonlinearly dispersive equations are little understood. Nevertheless it is obvious that the relationship between the 'scheme' dispersion and the dispersion of an approximate model is an important characteristic of a difference scheme for solving practical problems in wave hydrodynamics. As to the properties of difference schemes, especially if they are implemented on coarse grids [8], it would be appropriate to replace the historically formed hierarchy: 'a complete model — an approximate differential model — a discrete model' by the chain: 'the properties of a complete model — the properties of a discrete model that approximates an approximate model — the properties of an approximate differential model'. Our investigation consists of two stages. At the first stage we study the properties of nonlinearly dispersive models to construct a basis for computational algorithms. At the second stage we study the possibilities to construct the difference schemes whose properties would be consistent with those of the approximated models. Numerous numerical calculations show that the most important properties of the difference schemes besides the order of approximation and stability are associated with general * The work was supported by the Russian Foundation for the Basic Research (93-012-41). ' Computer Center, Siberian Branch of the Russian Academy of Sciences, Krasnoyarsk 660036, Russia 408 Z. 7. Fedotova and V. Yu . Pashkova concepts of mathematical modelling such as monotonicity, dissipativity, dispersion, conservativeness, invariance. Note that the method of differential approximation [10] is successfully employed for analysing the above properties as well as for constructing the difference schemes with given properties. In this work we have obtained the relationships between the 'scheme' dispersion and the dispersion of the approximated hydrodynamic model for a two-parameter family of finite difference schemes. We have found the ways, which are based on the method of the first differential approximation, to fit the numerical dispersion to the dispersion of the original differential model. We emphasize that in the work the differential models have been covered in some detail, but as to the efficiency of the numerical methods for these models only the first steps have been taken. 1. ON METHODS OF DERIVING NONLINEARLY DISPERSIVE MODELS Our approach is based on a comparative analysis and a hierarchy description of both numerous approximate models in hydrodynamics and the difference schemes that approximate them. As the first step in classification we have taken a model, which we call the 'base' model, because it is used for deriving many other models including the familiar ones. Our contribution to the investigation of the model is that we have considered it from the viewpoint of variational formulations (see [3,4]). We give the 'base' model and show techniques for obtaining approximate models in wave hydrodynamics. Recall that the long-wave models, when they are written in a dimensionless form, contain two small parameters: the nonlinearity parameter α = Q//I0 ^ ^ dispersion parameter β = (Λ0//0), where aQ is the characteristic wave amplitude, /IQ is the characteristic depth of a water basin, /Q is the characteristic wavelength [14]. When deriving the nonlinearly dispersive models we usually (it is not always the case) consider a potential flow of an inviscid incompressible fluid in a homogeneous gravity field. In this paper we consider the case of two space coordinates x, y in a horizontal direction and a vertical one, respectively. The bottom is considered to be impermeable and smooth: h = h(x) = /IQ = const. Then in dimensionless variables [14] this flow may be described by the Laplace equation for the velocity potential φ with nonlinear boundary conditions at the bottom and at the free surface and with the corresponding initial conditions for / = 0: = Ο , 0