Finite element method in Hermitian spaces in the analysis of irregular atmospheric processes

In the paper a verification of the difference schemes used to solve irregular initial value problems of the type of discontinuity surface and the interpretation of the results obtained will be presented. The problem of construction such methods, on the basis of finite differences, is closely connected with the improvement of the accuracy of numerical solutions in the neighborhood of a discontinuity point. The localization of such points is essentially simplified, if the difference method used enables us to reduce the width of deformation area of the discontinuity to a single space increment of the discrete grid. In such case one can use, for instance, procedures determining the maximum gradient of any sought-for function, which becomes in a certain region strong irregular or even discontinuous. This means that the maximum value of the first derivative of such a function must be determined one space increment of the grid. The finite element method in Hermitian space (function and its first derivative) seems to be one of the best suited for such a purpose.

Some difference schemes for the nonlinear equation (for example, a Korteweg - de Vries, tsunami) will be analyzed and some numerical solutions will be presented.