Difference Schemes for
Time-Dependent Equations with
Constant Coefficients
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 inves-
tigated 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 approx-
imations. 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.5.1 ONE-DIMENSIONAL HEAT CONDUCTION EQUATION
WITH CONSTANT COEFFICIENTSIn this section we consider the one-dimensional heat conduction equation
with constant coefficients and difference schemes in order to develop various
methods for designing the appropriate difference schernes in the case of
time-dependent problems.
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