For a function u(x,y,z,t) of three spatial variables (x,y,z) (see Cartesian coordinate system) and the time variable t, the heat equation is
∂u∂t−α(∂2u∂x2+∂2u∂y2+∂2u∂z2)=0
In the physical problem of temperature variation, u(x,y,z,t) is the temperature and α is the thermal diffusivity.
If you are used to determining numerical stability criteria for the 1D Fourier's equation, then almost identical methods may be used in 3D. Sine decompositions in all three directions are the way to proceed. For an implicit scheme, such as backward Euler, and for the equation you quoted, the method will be unconditionally stable.
Implicit time integration is used for the good stability properties.
A standard implicit Euler equation with second order central discretization is often used for this equation. It can be shown by the von Neumann analysis or by the Gershgorin eigenvalues locus that the scheme is uncoditionally stable. A second order time integration such as the Crank-Nicolson shows a very weak instability but this method is largely used, too.
Have a look here http://sgpwe.izt.uam.mx/files/users/uami/mlss/documentos/Morton.pdf
The Pure implicit or semi implicit schemes for above mentioned heat equation is always stable or we can say it is unconditionally stable. To check the stability cretirion, we can use any of the method from Von-Neumann analysis, Fourier method (in case of stable scheme) or Gershgorin’s principle.
http://www.ehu.eus/aitor/irakas/fin/apuntes/pde.pdf
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