Difference scheme for the Samarskii-Ionkin problem with a parameter (Q731576)

The paper deals with a weighted difference scheme for the problem \[ \frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}, \quad x \in (0,1) \] \[ u(0,t)=0, \quad \frac{\partial u}{\partial x}(0,t)+\frac{\partial u}{\partial x}(1,t)=0, \] \[ u(x,0)=u_0(x). \] Using the expansions in the explicit basis of eigenfunctions and associated functions of the non-selfadjoint spatial difference operator the authors find a necessary and sufficient stability condition with respect to the initial data in some energy norm which is equivalent to the grid \(L_2-\)norm. It is shown that this stability condition can not be weakened by choosing a different norm.