Excerpt from Stability Studies for Difference Equations: I Non-Linear Stability; II Coupled Sound and Heat Flow

In part II, the difference equations for coupled sound and heat flow, in which the latter is treated implicitly, are considered. The stability condition is known to be c at/ax l, where c is the isothermal sound speed. This seems somewhat paradoxical, for if the thermal conductivity is small, signals travel at practically the adiabatic sound speed, V7 0, which is larger than c. For such problems, stability, as usually defined, is inadequate for an actual calculation, since it considers only the limit At, Ax O. A practical stability criterion for such problems is proposed and is applied to the problem at hand, with complete success, as judged by numerical tests, which are also described; the criterion leads, in this problem, to a stability condition intermediate between c at/ax 1 and 'v7 c at/ax 1, depending on various parameters, includ ing At (or Ax) itself. The condition is shown to be necessary (by the Fourier analysis) and sufficient (by the energy method).

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