Stability of a family of difference schemes for the Samarskii-Ionkin problem with variable coefficient (Q2251859)

The author considers the heat equation \[ \frac{\partial u}{\partial t}=\frac{\partial}{\partial x}(k(x)\frac{\partial u}{\partial x})\qquad \text{ on } \qquad [0,T]\times[0,1] \] with boundary condition of the form \[ k(0)\frac{\partial u}{\partial x}(0,t)=k(1)\frac{\partial u}{\partial x}(1,t), \quad t>0. \] To approximate this problem at the point \( x_{i}\), a weighted difference scheme with parameter \(\sigma \) for the points \(t^{n}\) and \( t^{n+1}\) is used. The spectral properties of the difference operator of this scheme are used to establish the condition of uniform stability \[ \sigma \geq 0.5-h^{2}/4k_{2}\tau \] under the assumption \( 0< k_{1} \leq k(x)\leq k_{2}\).