Maximum Norm Stability of Difference Approximations to the Mixed Initial Boundary-Value Problem for the Heat Equation

We consider the heat equation ${u_t} = {u_{xx}}$ in the quarter-plane $x \geqq 0$, $t \geqq 0$ with initial condition $u(x,0) = f(x)$ and boundary condition $\alpha u(0,t) + {u_x}(0,t) = 0$. We are concerned with the stability of difference approximations ${\upsilon _\nu }^{n + 1} = Q{\upsilon _\nu }^n$ to this problem. Using the resolvent operator ${(Q - zI)^{ - 1}}$, we give sufficient conditions for consistent, onestep explicit schemes to be stable in the maximum norm.