Stability analysis of difference schemes for variable coefficient Schro¨dinger type equations

We consider the stability of difference schemes for the solution of the initial boundary value problem for the equation \[ u_1 = \left( {A(x,t)u_x } \right)_x + B(x,t)u + C(x,t)u + f(x,t)\], where u, A, B, C and f are complex valued functions. Using energy methods, we establish the stability of a general two level scheme which includes Euler’s method, Crank–Nicolson’s method and the backward Euler method. If the coefficient $A(x,t)$ is purely imaginary, the explicit Euler method is unconditionally unstable. For this case, we propose a new scheme with appropriately chosen artificial dissipation, which we prove to be conditionally stable.