Stability theory

In this chapter difference schemes for the simplest time-dependent equations are studied, namely, for the heat conduction equation with one or more spatial variables, the one-dimensional transfer equation and the equation of vibrations of a string. Two-layer and three-layer schemes are designed for the first, second and third boundary-value problems. Stability is investigated by different methods such as the method of separation of variables and the method of energy inequalities as well as by means of the maximum principle. Asymptotic stability of difference schemes is discovered for the heat conduction equation in ascertaining the viability of difference approximations. Finally, stability theory is being used, increasingly, to help us understand a variety of phenomena, so it seems worthwhile to discuss it in full details. 

Example. In order to apply the general stability theory for particular difference schemes, one needs to perform several manipulations ... 

The case of a skew-symmetric operator A. The main results of stability theory for two-layer schemes... 

Stability or instability of a scheme from the primary family depends only on selection rules for the operator R. From the point of view of stability theory the arbitrariness in the choice of the operator R is restricted by the following requirements ... 

Stability and convergence. The general stability theory for two-layer schemes applies equally well to the stability analysis of the weighted scheme (7). With this aim, the appropriate difference scheme with the homogeneous boundary conditions comes first ... 

VVe know from the general stability theory that scheme (10) is stable in the space Ha with respect to the initial data, that is,... 

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