A family of high-order accuracy explicit difference schemes with branching stability for solving 3-D parabolic partial differential equation (Q1841506)

A family of high-order accuracy \(O(\Delta t^2 + \Delta x^4)\) difference schemes for solving the 3-dimension parabolic equation \[ \frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+ \frac{\partial^2 u}{\partial z^2} \] is constructed. It needs to point out that at first such schemes of high-order accuracy were constructed and investigated by A.~Samarskij. The main result of the paper is that the stability condition is \[ r=\Delta t/\Delta x^2=\Delta t/\Delta y^2=\Delta t/\Delta z^2 < \tfrac 12. \] For the usual explicit scheme the similar condition restricts a time step heavier to \(r<1/6\). Nevertheless these schemes are not generalized to heat conduction equations with variable coefficients. That's why the obtained results are not of considerable interest for applied problems.