O metodach przybliżonego rozwiazywania problemów falowych w ośrodkach niesprężystych
of solution used are approximate methods, most often the methods of finite differences. In the case of utilizing the method of finite differences instead of a set of differential equations we solve a set of difference equations. The necessary condition for the convergence of an approximate solution to the exact solution is the stability of the set of difference equations. Chapter 2 contains the best
known definitions of stability for a linear set of difference equations with constant and variable coefficients.
The major purpose of this paper is to compare five methods of approximate solution of sets of partial differential equations of the first order of hyperbolic type with two independent variables. These are: the iteration method of Courant, method of direct integration, method of finite differences along the characteristics, the Keller-Thomée method and iteration methods which can be applied
only in the case of such sets of equations which can be reduced to a partial differential equation of the second order. These methods have been described in Chapters 3-7. Iteration methods and Keller-Thomée method ensure the convergence of the approximate solution to the exact solution of the differential equations. Courant's method, the method of direct integration and the method of finite differences along characteristics have been compared on the example of a cylindrical shear wave in an elastic/viscoplastic medium. In the case of Courant's method 3 iterations were calculated. The numerical results are characterized by good conformity.
The problem of stability of the methods considered has been discussed. The iteration methods are analytical methods and for them the problem does not exist. For the Keller-Thomée method the absolute stability has been demonstrated in [28]. For the remaining two methods the stability problem is open. The stability of these two methods has been analyzed in the case of solving a set of two semi-linear Eqs. (8.1), describing the problem of propagation of a cylindrical shear wave in an elastic/viscoplastic medium. Taken there as a basis were the papers [23] and [55]. It has been shown that in the case of the method of finite differences along the characteristics the definition of stability described in Chapter 2 cannot be utilized.
Chapter 10 contains some considerations concerning the continuity of derivatives of the solution for the cylindrical shear wave in an clastic/viscoplastic on a load curve,
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