Numerical Stability Condition for Heat Transfer Simulation of Superfluid Helium

Numerical stability of the heat transfer equation of superfluid helium is discussed in detail for one- and two-dimensional cases. From the viewpoint of diffusive property of heat conduction, the stability conditions for explicit finite difference equations describing the so-called 1/3-power law have been derived. The stability condition depends on the temperature gradient as well as the heat conductivity function and the mesh spacing. To maintain the numerical stability, the time step should be lowered for low temperature gradients. A linearization of the 1/3-power law for low temperature gradients is useful to suppress the numerical instability, but a threshold of the linearization should be selected as low as possible not to affect numerical results. Finally, the validity of the stability conditions is demonstrated by performing two-dimensional numerical simulations of the natural convection of liquid helium with λ-transition.