For initial boundary value problems (in the first and second approximation) of the shallow-water theory we consider their mathematical statement and finite difference methods of solving them on nonstationary adaptive grids. We show the advantage of adaptive grids constructed by the equidistribution method over stationary ones when numerically modelling the run-up of wave on the wall.

on the interaction of a solitary wave of amplitude al with a plane vertical wall on which the wave is incident at angle !bl. It is established that, depending on the value of ~bi, the reflection of the wave from the wall can be regular or irregular (Mach reflection). In regular reflection, the crests of the incident and reflected waves intersect on the wall, and in Mach reflection, a third wave, called the Mach stem, appears between the wall and the point of intersection between the crests of the first two waves. Mach reflection is shown schematically in Fig. 1, where the wave crests are indicated by heavy lines.

We consider the mathematical statement and finite difference methods of solving the twodimensional initial boundary value problems in the first approximation in the shallow-water theory, using nonstationary adaptive grids. We give the examples of modelling numerically the wave processes in a basin with curvilinear boundary.

Abstract. Numerical research results of deciduous tree ignition by cloud-to-ground lightning discharge are presented. The problem is solved in one-dimensional statement in cylindrical system of coordinates. The typical range of influence parameters of positive and negative cloud-to-ground lightning discharges is considered. Ignition conditions for deciduous tree are established.

We consider the finite difference algorithm for modelling surface waves in the framework of one nonlinear dispersive model. The algorithm proposed allows us to use nonstationary adaptive grids adjusting to the complex geometry of the domain and to the peculiarities of the solution. A distinctive feature of the algorithm is to separate the elliptic and hyperbolic parts in the original equations. In order to solve an elliptic equation obtained we construct a finite difference approximation of the scheme of the type Oblique cross' with a selfadjoint and positive definite operator. We estimate the boundaries of the spectrum of this operator. We give an example of the numerical modelling of the oblique run-up of a solitary wave on a vertical wall.